iS6 


UNIVERSITY    OF    CALIFORNIA 


DEPA'^TMFNIT-   nc    cm  lr^A-^Ir^M 


No.  /a 


THE   PUBLIC   SCHOOL   ARITHMETIC 

FOR  GRAMMAR  ORADES 


•y^y^' 


THE 

PUBLIC  SCHOOL  ARITHMETIC 

FOR  GRAMMAR  GRADES 

BASED  ON   McLELLAN  AND  DEWEY'S 

"PSYCHOLOGY    OF    NUMBER" 

BY 

J.   A.   McLELLAN,  A.M.,   LL.D. 

PRESIDENT  OF  THE  ONTARIO  NORMAL.  COLLEGE  ;   AUTHOR  (WITH  DR.  DEWEY) 

OF  "the  PSYCHOLOGY  OF  NUMBER,"   "APPLIED  PSYCHOLOGY," 

"  THE  teacher's  HANDBOOK  OF  ALGEBRA,"   ETC. 

AND 

A.   F.   AMES,   A.B. 

HONOR  GRADUATE  IN  MATHEMATICS;   FORMERLY  MATHEMATICAL  MASTER 

ST.   THOMAS  COLLEGIATE   INSTITUTE,   ETC.  ;   SUPERINTENDENT 

OF  SCHOOLS,  RIVERSIDE,  ILL. 


THE   MACMILLAN   COMPANY 

LONDON:    MACMILLAN  &  CO.,  Ltd. 
1902 

All  rights  reserved 


Copyright,  1902, 
By  the  MACMILLAN  COMPANY. 

EDUCATIOra  DEFT* 


J.  S.  CushinK  &  Co.  —  Berwick  &  Smith 
Norwood  Mut.  U.  S.  A. 


/  r 


PREFACE 

This  book  has  been  specially  prepared  for  use  in  Grammar 
Grades.  Like  all  the  other  books  of  the  series,  the  treatment  is 
based  on  Dewey  and  McLellan's  "  Psychology  of  Number  "  ;  the 
basal  idea  of  that  work  is  the  correlation  of  number  and  measure- 
ment —  "^o  number  without  measurement,  no  measurement  without 
number.""  This  implies  three  factors  in  the  number  process,  —  a 
Whole  to  be  measured,  a  Unit  of  measure,  and  the  How  Many  of 
this  Unit  in  the  Whole.  This  basal  idea  is  nothing  but  an  appli- 
cation to  quantity  of  the  fundamental  principle  common  to  all 
thinking :  — 

From  undefined  whole  to  defined  whole  by  Analysis,  which  gives 
parts  (or  particulars),  and  by  Synthesis,  which  gives  the  defined  whole. 

The  application  of  this  principle  in  the  teaching  of  Arithmetic 
makes  all  the  difference  between  the  rational  or  psychological 
method  and  the  baldly  mechanical  methods  which  too  generally 
prevail.  This  one  idea,  the  idea  of  number  as  measurement,  run- 
ning through  all  the  so-called  rules  of  Arithmetic,  secures  the 
unity  of  the  whole  subject,  and  maintains  thereby  the  constantly 
increasing  interest  of  the  learner.  The  mind  naturally  works 
toward  unity,  and  anything  that  facilitates  this  normal  move- 
ment must  be  essentially  attractive,  while  anything  that  thwarts 
it  is  essentially  repellent. 

The  lack  of  recognition  of  this  principle  largely  explains  the 
distaste  for  Arithmetic  which  may  be  found  in  so  many  schools. 
Each  rule  is  practically  regarded  as  a  bit  of  subject-matter  which 
somehow  has  to  be  got  into  the  child's  mind  independently  of  his 
previous  experiences  in  arithmetical  ideas  and  processes.  When 
e.g.  teachers  and  text-book  writers  look  upon  Fractions  as  having 

544r}«5 


'vr  ....--.  PREFACE 

iiS  ccmitectto'rr  "^ttlx  "'Whole  Numbers,"  as,  indeed,  not  even  to  be 
classed  as  numbers,  is  it  any  wonder  that  the  child,  when  he  comes 
to  fractions,  is  utterly  bewildered,  separated  as  he  is  from  his 
former  number  experiences  by  a  break  which  he  cannot  pass  ? 
For  the  child  learns  with  what  he  has  learned.  When  any  new 
matter  is  presented  to  him,  there  must  be  a  breaking  up  into 
ideas  and  images  of  all  that  part  of  what  "  he  has  learned  "  which 
is  felt  to  have  a  bearing  upon  the  new  matter.  The  selection  and 
adjustment  of  these  ideas  is  the  instrument  by  which  the  new  is 
learned  ;  that  is,  by  which  it  is  interpreted  and  assimilated.  The 
result,  in  fact,  is  a  remaking  of  the  old  —  the  enlarging,  defining, 
and  enriching  of  the  old  experiences  by  means  of  the  new.  It  is 
plain,  then,  that  if  there  is  nothing  in  the  old  experience  con- 
nected with  the  new  matter,  or  if  the  old  experience  is  only  very 
vaguely  connected  with  it,  learning  in  the  true  sense  of  the  word 
cannot  take  place. 

On  the  other  hand,  the  psychological  method,  beginning  with 
the  vague  How  Much  and  How  Many  of  the  child's  experiences 
and  proceeding  by  means  of  constructive  exercises  in  which  the 
need  of  number  is  felt,  leads  to  a  clear  idea  of  number  as  a  defi- 
nite So  Many  of  units  of  measure  making  up  the  definite  So 
Much  of  quantity.  There  is  unity  from  beginning  to  end.  Each 
successive  lesson  grows  out  of  the  preceding  one  and  contributes 
to  the  development  of  the  number  sense.  The  rules  are  not 
isolated  groups  of  ideas  and  operations  having  no  intelligible  con- 
nection with  one  another,  but  are  simply  phases  in  the  develop- 
ment of  the  one  idea  which  the  child  has  been  using  from  the 
start.  The  "  Images  "  which  come  from  his  first  crude  notions  of 
number  and  quantity  are  the  means  for  the  mastery  of  rational 
counting ;  the  images  which  result  from  adding  and  subtracting 
are  easily  used  for  the  recognition  of  the  new  element  in  mul- 
tiplication and  division  —  new  only  in  this,  that  the  ratio  idea, 
implicitly  present  from  the  first,  becomes  more  explicit. 

The  same  is  true  of  Fractions.  Fractions  are  not  something 
wholly  outside  the  child's  experience.  He  has  been  unconsciously 
using  the  fraction  process  from  the  first.     He  has  been  breaking 


PREFACE  Vll 

up  a  whole  into  parts  and  putting  the  parts  together  again,  e.g, 
he  has  measured  a  length  of  eight  inches  and  counted  the  inch 
divisions.  In  this  counting  he  is  taking  one  of  the  eight,  two  of 
the  eight,  three  of  the  eight,  etc.  This  is  the  fundamental  process 
of  fractions.  It  is,  too,  essential  to  the  growth  of  the  right  idea 
of  number. 

When  the  method  of  text-book  and  teacher  is  based  on  a  clear 
conception  of  the  mind's  action  in  dealing  with  quantity,  there 
will  be  no  need  for  elaborate  controversy  as  to  whether  the  course 
in  arithmetic  should  be  simplified,  or,  rather,  impoverished,  by 
leaving  out  certain  "  Rules  "  —  Least  Common  Multiple,  Percent- 
age, Interest,  Ratio  and  Proportion,  etc. 

Some  of  the  features  of  this  book  may  be  more  specifically 
stated :  — 

1.  The  treatment  of  the  subject  being  based  on  the  true  idea 
of  number  —  that  idea  of  number  which  brings  the  world  into 
subjection  to  man  —  the  unity  of  the  subject-matter  is  deter- 
mined ;  the  unity  of  the  subject-matter  secures  unity  of  interest 
and  the  normal  action  of  the  mind  from  beginning  to  end.  This 
means  the  highest  results  with  the  least  waste  of  power. 

2.  This  book,  therefore,  makes  for  thoroughness.  Most  arith- 
metics aim  at  attaining  thoroughness  by  repetition  ;  but  repetition 
is  not  thoroughness,  nor  does  it  lead  to  thoroughness  ;  on  the  con- 
trary, it  may  be  a  hindrance  to  thoroughness.  Endless  repetition 
of  forms  and  symbols,  whether  by  the  Grube,  the  Spiral,  or  any 
other  method,  where  there  is  little  or  no  clear  consciousness  of 
the  meaning  of  symbols,  or  the  mental  activity  symbolized,  tends 
to  dulness  and  mind-wandering  instead  of  alertness  and  attention. 
Not  repetition  of  facts  and  principles,  but  unity  of  thought,  quick- 
ened by  unity  of  interest  and  working  upon  facts  and  principles 
to  organize  them  into  unity,  is  thoroughness ;  or,  as  Dr.  Dewey 
somewhere  says,  "  One  is  thorough  who  pursues  a  given  portion 
of  work  consecutively  and  adapts  the  various  parts  of  it  to  each 
other  in  an  effective  way  so  as  to  make  a  whole." 

3.  With  this  idea  of  thoroughness  constantly  in  view,  all  the 
explanations  and  exercises  are  so  arranged  that  there  is  a  contin- 


VIU  PREFACE 

nous  transformation  of  attainments  into  resources.  When  a  new 
problem,  new  thought,  or  topic  is  "  learned  '^  by  means  of  relevant 
images  derived  from  past  experiences,  it  is  immediately  applied 
in  the  acquisition  of  new  knowledge.  There  is  thus  that  conti- 
nuity which  alone  secures  organized  knowledge  and  organized 
power. 

4.  In  its  vast  number  of  new  problems  founded  on  and  grow- 
ing out  of  the  commercial,  agricultural,  industrial,  and  social 
interests  of  the  country,  this  book  is  without  a  rival.  These 
problems  are  so  presented  that  the  pupil  can  hardly  help  realiz- 
ing the  actual  conditions  which  gave  rise  to  them,  or  to  which 
they  allude ;  the  result  is  a  vital  connection  between  the  life  of 
the  school  and  the  larger  social  life.  They  are  therefore  practical 
in  the  true  sense  of  the  word.  They  make  possible  the  true  cor- 
relation of  Arithmetic,  viz.,  with  the  activities  of  the  child  and  of 
the  community  in  which  he  lives.  The  teacher  can  depend  on  the 
accuracy  of  the  figures  and  statements  given  in  the  problems. 

5.  On  account  of  the  careful  arrangement  of  the  problems  with 
a  view  to  the  gradual  and  normal  growth  of  the  image,  the  great 
instrument  of  the  teaching  as  well  as  of  the  learning  processes, 
the  book  is  at  once  stimulating  and  easy  to  teach.  The  book 
teaches  itself.  There  is,  and  should  be,  no  break  between  the 
subject-matter  and  method. 

6.  The  miscellaneous  exercise  at  the  close  of  nearly  every 
chapter  gives  ample  opportunity  for  review,  aiid  will  prove  a 
great  saving  of  time  to  the  teacher. 

7.  Abstract  problems  and  principles  are  constantly  and  closely 
associated  with  concrete  examples,  that  is,  with  ideas  and  images 
derived  from  the  learner's  own  experience. 

8.  The  ratio  idea  is  used  in  a  sane,  practical  way  throughout 
the  book.  No  number  without  measurement  and  no  measure- 
ment without  ratio. 

9.  There  is  a  gradual  and  psychological  introduction  of  "  X " 
at  the  time  when  the  pupil  feels  the  need  of  a  more  powerful  and 
economical  instrument  of  solution  than  is  afforded  by  arithmetic. 
The  "X"  symbolizes,  gathers  into  one  the  elements  which  are 


PREFACE  ix 

more  or  less  distinctly  seen  to  be  connected  with  the  problems, 
and  thus  facilitates  the  calling  up  of  the  necessary  images.  This 
furnishes  a  real  basis  and  motive  for  the  study  of  algebra. 

10.  There  is  an  introduction  to  concrete  geometry  in  a  series 
of  closely  related  problems. 

■  This  series  of  "  Public  School  Arithmetics "  is  the  only  series 
founded  upon  the  "Psychology  of  Number,"  and  from  the  first 
book  to  the  last  consistently  carrying  out  the  fundamental  ideas 
of  that  work.  The  many  teachers  using  the  series  are  unanimous 
in  expressing  a  high  opinion  of  it.  It  is  thought  that  the  words 
which  Dr.  Dewey  applied  to  "The  Public  School  Arithmetic" 
may  be  applied  with  emphasis  to  this  "  Public  School  Arithmetic 
for  Grammar  Grades  "  :  — 

"  I  am  particularly  struck  with  the  clearness  and  conciseness 
of  the  method  of  treatment,  the  logical  order  of  the  selection  of 
topics,  and  the  exclusion  of  useless  and  irrelevant  matter.  The 
simplification  of  treatment,  due  to  sticking  close  to  fundamental 
principles,  must  recommend  the  book  to  teachers  and  pupils  who 
have  been  bewildered  by  the  great  number  of  topics  treated  in 
the  ordinary  arithmetics  —  topics  which  do  not  differ  at  all  in 
their  logical  or  arithmetical  basis,  but  are  simply  different  prac- 
tical expressions  of  the  same  principle." 

An  answer  book  will  be  furnished  to  classes,  free  of  charge,  on 
the  written  order  of  the  teacher. 


CONTENTS 


CHAPTER  I 

PAGE 

Definitions  and  Review 1 

CHAPTER  II 
Review      , .14 

CHAPTER  III 
Numeration  and  Notation 28 

CHAPTER  IV 
Addition 35 

CHAPTER  V 
Subtraction ,        .50 

CHAPTER  VI 
Multiplication 66 

CHAPTER  VII 
Division 84 

CHAPTER  VIII 
Comparison  of  Numbers        ........    103 

CHAPTER  IX 
Square  Root ,    106 

CHAPTER  X 

Greatest  Common  Measure  and  Least  Common  Multiple  .    117 

xi 


xii  CONTENTS 

CHAPTER  XI 

PAOB 

Fractions  . 124 

CHAPTER  Xn 
Decimals 171 

CHAPTER  Xni 
Compound  Quantities     .        .        .        .        .        .        .        ,        .     193 

CHAPTER  XIV 
Percentage 234 

CHAPTER  XV 
Interest 272 

CHAPTER  XVI 
Ratio  and  Proportion  .        .        .        ...        .        .        .    287 

CHAPTER  XVII 
Square  Root    . .        .        .296 

CHAPTER  XVIII 
Mensxjration    ...        .        . 298 

CHAPTER  XIX 
Metric  System 320 

CHAPTER  XX 
Miscellaneous  Exercise 328 

-       CHAPTER  XXI 
Appendix  .        .        .        . 349 


THE   PUBLIC   SCHOOL   ARITHMETIC 

FOR  GRAMMAR  GRADES 


ARITHMETIC 

CHAPTER  I 

DEFINITIONS   AND   REVIEW 

1.  A  unit  is  a  quantity  used  to  measure  quantity  of  the 
same  kind. 

Thus,  1  mi.  is  a  unit  used  to  measure  distance,  1  A.  to  measure  the  size 
of  a  farm,  and  1  doz.  eggs  to  measure  the  quantity  of  eggs.  If  a  lady  wants 
to  know  into  how  many  liair  ribbons,  each  1  ft.  6  in.  long,  she  can  cut  a  piece 
of  ribbon  2  yd.  1  ft.  6  in.  long,  she  must  use  1  ft.  6  in.  as  a  unit  with  which 
to  measure  the  length  2  yd.  1  ft.  6  in.  Cut  a  piece  of  string  2  yd.  1  ft.  6  in. 
long  and  divide  it  into  parts  each  ]  ft.  6  in.  long.     How  many  ? 

2.  As  a  person  measures  a  quantity  he  counts  the  Number 
of  units  in  the  quantity.  The  number  and  the  unit  measure 
the  quantity. 

Draw  a  line  1\  in.  long  and  measure  it  with  a  unit  2|  in.  How  many 
2|-in.  units  in  7|  in.  ?  Measure  the  same  quantity  with  a  unit  1|  in.  How 
many  \\-m.  units  in  1\  in.  ? 

3.  The  number  is  the  Ratio  of  the  quantity  to  the  unit. 

A  line  15  in.  long  is  measured  5  times  by  the  unit  3  in.,  hence  5  is  the 
ratio  of  15  in.  to  3  in.  What  is  the  ratio  of  6  in.  to  2  in.  ?  12  qt.  to  3  qt.  ? 
1  hr.  to  10  min.  ?     1  ft.  6  in.  to  6  in.  ? 

.  Exercise  1 

1.  Mark  two  points  on  the  blackboard  6  yd.  apart.  Cut  a 
string  2  ft.  long,  and  by  measuring  find  how  often  2  ft.  is  con- 
tained in  6  yd. 

B  1 


2  ARITHMETIC 

2-  iFind,  without  actually  measuring,  how  often  2  ft.  is  con- 
tainecl  in  6  yd.  How  many  widths  of  carpet  2  ft.  wide  are  needed 
to  carpet  a  room  G  yd.  wide  ? 

3.  Draw  an  oblong  12  in.  long  and  8  in.  wide.  Cut  out  of 
paper  an  oblong  4  in.  long  and  3  in.  wide.  Using  this  as  a  unit 
of  measure,  find  how  often  it  is  contained  in  the  larger  oblong. 

4.  Find,  without  actually  measuring,  how  many  oblongs  4  in, 
by  3  in.  can  be  cut  from  an  oblong  12  in.  by  8  in. 

5.  A  piece  of  cardboard  18  in.  by  12  in.  is  cut  into  cards  6  in. 
long  and  4  in.  wide.     How  many  are  there  ? 

6.  Put  2  gal.  1  qt.  of  water  into  a  pail  and  measure  it  with  a 
pint  measure.     2  gal.  1  qt.  =  ?  pt. 

7.  Find,  without  actually  measuring,  how  many  pints  are 
equal  to  2  gal.  1  qt. 

8.  How  many  pint  jars  can  be  filled  with  4  gal.  2  qt.  of  maple 
syrup  ?     ■' 

9.  Cut  out  a  piece  of  cardboard  4  in.  long.  Measure  a  dis- 
tance equal  to  8  of  these  units.  Measure  this  distance  again  with 
a  foot  rule.     8  times  4  in.  =  ?  ft.     ?  in. 

10.  Find,  without  actually  measuring,  how  many  feet  and 
inches  a  string  is  that  can  be  cut  into  8  pieces  each  4.  in.  long. 

11.  In  the  following  examples  what  number  expresses  the 
measurement  of  the  quantity  by  the  unit? 

Quantity  Unit  Quantity  Unit 

2  ft.  6  in.                3  in.  3  bu.  12  qt. 
11  qt.  2  pt.              6  pt.  $4.50  1  dime 

3  wk.  4  da.              5  da.  24  sq.  in.  oblong  3  in.  x  2  in. 
1  hr.  30  min.  18  min.  30^  3^  +  2^ 

How  do  you  find  th^  number  of  units  in  a  given  quantity  ? 

12.  In  the  preceding  example  what  is  the  ratio  of  each  quan- 
tity to  its  own  unit  ?  What  is  the  ratio  of  the  unit  to  the 
quantity  ? 


DEFINITIONS  AND  KEVIEW  8 

13.  In  the  following  examples  find  the  quantities  measured  by 
the  numbers  and  the  units: 

Number                              Unit  Number                    Unit 

8                           2qt.  6  SiovlO^ 

2|                        6^  9  2pk. 

3  35  mi.  40  5  da. 

2  doi;.                   2  ^  apiece  .6  3  mi.  +  4  mi. 

14.  How  do  you  find  the  quantity  containing  a  given  number 
of  units  ? 

Exercise  2 

1.  A  boy  saves  $  4  a  month.  In  how  many  months  will  he  save 
the  price  of  a  bicycle  worth  $  32  ? 

2.  A  man  earns  f  18  a  week  and  spends  $  15.  In  how  many 
weeks  will  he  save  enough  to  pay  a  debt  of  $  24  ? 

3.  A  family  uses  21  qt.  of  milk  at  6  ^  a  quart  daily.  What 
does  their  milk  cost  per  day  ? 

What  do  they  pay  for  milk  in  a  month  of  30  da.  ? 

4.  A  family  uses  1  qt.  1  pt.  of  milk  at  6  ^  a  quart  daily. 
What  is  their  milk  bill  for  the  month  of  April  ?     May  ? 

5.  A  family  uses  2  qt.  1  pt.  of  milk  at  6  ^  a  quart  daily. 
AVhat  do  they  pay  for  milk  in  the  month  of  March,  if  they  get 
8  qt.  extra  ? 

6.  A  boy  riding  on  a  bicycle  gains  3  mi.  an  hour  on  a  man  who 
is  driving.  In  how  many  hours  after  passing  him  will  he  be 
15  mi.  ahead  ? 

m  7.  Two  boats  travel  down  a  river  at  the  rate  of  14  and  9  mi. 
an  hour  respectively.  If  they  start  together,  when  will  the  faster 
be  20  mi.  ahead  ? 

8.  Two  trains  travel,  one  east  and  the  other  west,  at  the 
respective  rates  of  35  and  23  mi.  an  hour.  How  far  apart  will  the"'' 
be  in  3  hr.  ? 


4  ARITHMETIC 

9.    How  do  you  find  the  area  of  an  oblong  ? 
Find  the  cost  of  cementing  the  floor  of  a  cellar  8  yd.  long  and 
6  yd.  wide  at  15  ^  a  square  yard. 

10.  Find  the  number  of  cubic  feet  in  a  solid  4  ft.  long,  3  ft. 
wide,  and  2  ft.  thick.  If  1  cu.  ft.  weighs  8  lb.,  find  the  weight  of 
the  solid. 

11.  Find  the  cost  of  digging  a  cellar  6  yd.  long,  4  yd.  wide, 
2  yd.  deep  at  24  ^  a  cubic  yard. 

Exercise  3 

In  the  following  examples  find  the  sum  and  prove  the  answers 
correct  by  beginning  at  the  top  and  adding  down : 

1.    165  2.    913  3.   319  4.   231 

952  567  495  908 

674  846  323  259 


5.  1829 

6.  1173 

7. 

2161 

8.  6963 

6513 

2664 

7243 

8196 

8661 

3849 

5436 

7749 

9.  66981 

10.  65534 

11. 

36205 

12.  90195 

46938 

23776 

18882 

89974 

33430 

50388 

68804 

36632 

22109 

72441 

15234 

88353 

.3.  $2581.27 

14. 

$8394.20 

15.  $5758.93 

2915.74 

7947.38 

2392.34 

187.29 

4698.63 

1332.55 

3433.75 

6254.98 

4671.12 

2869.28 

6682.57 

6778.86 

6.  8667.914 

17. 

5639.498 

18.  2542.368 

2336.621 

4880.436 

4923.552 

1138.325 

2252.203 

1589.865 

3657.562 

6767.859 

3656.146 

6551.438 

8599.638 

2354.289 

DEFINITIONS  AND  REVIEW 


Exercise  4 


1.  An  ocean  steamer  carried  627  cabin  and  829  steerage  pas- 
sengers.    Find  the  total  number  of  passengers. 

2.  A  vessel  in  crossing  the  Atlantic  made  daily  runs  of  496, 
443,  470,  457,  481,  and  425  mi.     Find  the  length  of  the  trip. 

3.  In  June,  1899,  a  rural  mail-carrier  delivered  5089  pieces 
of  mail ;  in  July,  5456 ;  August,  5942 ;  September,  6094 ;  October, 
6799.  Find  the  total  number  of  pieces  delivered  in  the  five 
months. 

4.  In  a  game  of  golf  a  gentleman  took  the  following  number 
of  strokes  at  the  different  holes :  6,  8,  8,  7,  5,  7,  4,  6,  8.  F'ind  his 
score. 

5.  Aug.  16,  1899,  a  Chicago  daily  paper  announced  the 
following  contributions  to  the  Porto  Rican  relief  fund:  $500, 
flOO,  $125,  $1,  $10,  $2,  $5,  $105.     Find  the  total. 

6.  The  following  cash  receipts  of  the  Daily  News  Fresh  Air 
Fund  were  reported  Aug.  5,  1899:  $.25,  $.10,  $9.75,  $2.00, 
$3.45,  $1.00,  $10.00,  $1.50,  $1.80,  $.50,  $2.75,  $1.00.  Find 
the  total  amount. 

7.  The  daily  attendance  at  the  Lincoln  Park  Sanitarium  for 
the  week  ending  Saturday,  Aug.  19,  1900,  is  given  below.  Find 
the  total  attendance  for  each  day  of  the  week. 

Monday,  Aug.  14.  —  Sick  babies,  157;  mothers,  157;  children,  ^17; 
visitors,  1044. 

Tuesday,  Aug.  15. — Sick  babies,  153;  mothers,  153;  children,  696; 
visitors,  1026. 

Wednesday,  Aug.  16. — Sick  babies,  158  ;  mothers,  159;  children,  846  ; 
visitors,  1210. 

Thursday,  Aug.  17. —Sick  babies,  211;  mothers,  212;  children,  1362; 
visitors,  2062. 

Friday,  Aug.  18.  —  Sick  babies,  145  ;  mothers,  145  ;  children,  730  ; 
visitors,  1086. 

Saturday,  Aug.  19. —Sick  babies,  158;  mothers,  158;  children,  637; 
visitors,  1107. 


6 


ARITHMETIC 


8.  Find  the  total  number  of  recruits  enlisted  for  service  in 
the  Philippine  Islands,  Sept.  13,  1899,  the  record  of  the  different 
regiments  on  that  date  being  as  follows :  38th,  650 ;  39th,  800 ; 
40th,  262;  41st,  237;  42d,  484;  43d,  304;  44th,  311 ;  45th,  448; 
46th,  514  ;  and  47th,  592. 

9.  Four  towns  lie  on  a  road  running  north  and  south.  The 
distance  from  the  first  to  the  second  is  6.79  mi.,  from  the  second 
to  the  third  8.57  mi.,  and  from  the  third  to  the  fourth  9.84  mi. 
Find  the  distance  between  the  first  and  fourth  towns. 

10.  Aug.  17,  1899,  there  were  inspected  in  Chicago  132  more 
cars  of  oats  than  of  corn.  If  281  cars  of  corn  were  inspected,  find 
the  number  of  cars  of  both  kinds  inspected  on  that  day. 

11.  A  country  post-office  received  746  pieces  of  mail  matter  in 
August,  144  more  in  September  than  in  August,  and  40  more  in 
October  than  in  September.  Find  the  total  number  of  pieces 
received  during  these  three  months. 

12.  A  real  estate  agent  bought  three  lots  for  $375,  $490,  and 
$550  respectively.  He  sold  them  at  a  gain  of  $125  apiece. 
Find  the  total  selling  price. 

13.  A  man  paid  $  3475  for  a  house  and  lot.  He  spent  $  625 
for  improvements  and  $  35  in  taxes.  He  then  sold  at  a  gain  of 
$  450.     Find  the  selling  price. 

14.  Copy  neatly  the  following  statement  of  six  weeks'  cash 
receipts;  add  the  amounts  vertically  and  find  the  sum  of  the 
totals : 


MON. 

TUES. 

Wed. 

TllUE. 

Fri. 

Sat. 

1st 

$32.25 

$24.63 

$  25.93 

$  36.27 

$23.92 

$46.76 

2d 

41.38 

18.79 

45.10 

24.50 

33.37 

37.89 

3d 

29.50 

31.83 

20.48 

39.75 

36.76 

36.97 

4th 

27.68 

29.66 

19.75 

18.42 

86.61 

26.61 

5th 

38.67 

42.77 

27.86 

31.20 

47.63 

47.38 

6th 

28.34 

28.64 

19.99 

26.46 

29.75 

26.74 

DEFINITIONS  AND  PEVIEW 

15.    Copy  and  add  as  in  tlie  preceding  example : 


1st 

2d 

3d 

4th 

6th 

6th 

Mon. 

$32.25 

$41.38 

$  29. 50 

$27.68 

$38.67 

$28.34 

Tiies. 

24.63 

18.79 

31.83 

29.65 

42.77 

28.64 

Wed. 

25.93 

45.10 

26.48 

19.75 

27.86 

19.99 

Thur. 

36.27 

24.50 

39.75 

18.42 

31.20 

26.45 

Fri. 

23.92 

33.37 

36.76 

36.61 

47.53 

29.75 

Sat. 

46.75 

37.89 

36.97 

26.51 

47.38 

25.74 

Why  should  you  get  the  same  total  amount  for  this  example  as 
for  the  previous  one  ? 

16.  Find  the  total  cost  of  removing  from  the  business  district 
of  the  city  of  Chicago  the  snow  that  fell  during  the  week  ending 
Feb.  9,  1901. 


5torm  No.  1, 
St  of  laborers     . 
st  of  teams    . 

.     .     .     $8,950 
.     .     .      11,710 

Storm  No.  2, 
Cost  of  laborers      . 

r!nsf,  of  t.p.n.Tns 

.    .    .    $3,300 
.     .     .        4,200 

*  Subtract: 

Exercise  5 

1.  833 

752 

2.   527 
481 

3. 

496 
169 

4.   943 
755 

5.   1931 
1685 

6.    6743 

2862 

7. 

9236 
7698 

8.    7693 
2498 

9.   65954 

58712 

10.    47581 
29649 

11. 

38145 
24628 

12.    99570 
41657 

13.   2678.28 
1732.50 

14. 

3650.12 
1732.43 

15.    8418.60 
1654.75 

16.   1559.834^ 
263.761 

17. 

3976.986 
1932.578 

18.   5436.621 
1864.865 

* 

Use  the  addition  method  of  subtracting. 

8  ARITHMETIC 

19.  2874.931         20.  9025.723        21.  3542.790 
1458.923  3496.824  1863.254 

22.  408025.3         23.  53251.04        24.  3868.239 
372486.7  21740.42  2687.832 

Exercise  6 

1.  Aug.  21,  1899,  the  highest  temperature  in  Chicago  was  90° 
and  the  lowest  74°.     Find  the  difference. 

2.  Of  1310  men  composing  the  30th  regiment  706  were  from 
Illinois  and  the  remainder  from  Michigan.  Find  how  many  were 
from  Michigan.  How  many  more  were  from  Illinois  than  from 
Michigan  ? 

3.  The  freight  earnings  of  a  certain  railroad  per  mile  for  1899 
were  $4675,  and  for  1898  $4138.     Find  the  increase. 

4.  August,  1899,  the  po.stal  receipts  at  St.  Paul,  Minn., 
amounted  to  $37,804,  an  increase  of  $255  over  August,  1898. 
Find  the  postal  receipts  at  St.  Paul,  August,  1898. 

5.  The  cost  of  police  service  in  a  certain  town  for  1898  was 
$  3759.16  and  for  1899  $  4263.74.     Find  the  increase. 

6.  A  town  board  levied  for  police  $3648.73,  for  a  street  fund 
$  1682.79,  and  for  a  fire  and  water  fund  $  3763.62.  The  tax  for 
these  purposes  in  the  preceding  year  was  $7854.62.  Find  the 
increase. 

7.  The  report  of  the  Chicago  Penny  Savings  Society  shows 
that  the  deposits  for  the  year  ending  June  30, 1899,  were  $43,300, 
and  the  withdrawals  $  31,849.75.  Find  the  balance  to  the  credit 
of  depositors  June  30,  1899. 

8.  The  area  of  the  Philippine  Islands  is  114,326  sq.  mi.,  of 
Cuba  41,655  sq.  mi.,  and  of  Porto  Rico  3550  sq.  im.  How  much 
larger  are  the  Philippine  Islands  than  Cuba  and  Porto  Rico 
together  ? 


DEFINITIONS  AND  REVIEW  9 

9.  June  30,  1900,  a  school  board  had  $1552.67  on  hand. 
During  the  next  year  it  received  $8795.66  and  spent  $8973.54. 
Find  the  balance  on  hand  June  30,  1901. 

10.  In  1879  the  total  length  of  our  postal  routes  was  79,991  mi., 
in  1898  it  was  174,777  mi.     Find  the  increase. 

11.  The  exports  from  the  United  States  to  the  Philippines 
from  Jan.  1  to  July  31,  1899,  were  $386,109,  for  the  same  period 
in  1898  $  65,736,  and  in  1897  $  47,754.  Find  the  increase  in  the 
amount  of  exports  each  year. 

12.  In  1899  the  Chicago,  Milwaukee,  and  St.  Paul  Eailroad 
owned  7876.84  mi.  of  track,  of  which  6142.31  mi.  were  main  line. 
Find  the  number  of  miles  in  the  branch  lines. 

Exercise  7 

*  Multiply  and  prove  the  answers  to  the  first  two,  correct  by 
multiplying  the  multiplier  by  the  multiplicand. 


1. 

61 
17 

96 
35 

34 
56 

83 

98 

35 

75 

66 
47 

2. 

342 

86 

190 
19 

238 
67 

652 
45 

385 

98 

793 

57 

3. 

$  67.40 

85 

$  72.59 
67 

$23.55 
92 

$  97.64 
96 

$  84.07 
45 

4. 

$  64.92 
38 

$  87.96 

75 

$53.28 
36 

$  93.16 

77 

$  80.57 
99 

5. 

601 
324 

141 

188 

248 
312 

884 
144 

345 
432 

6. 

1491 
834 

2983 
402 

1504 
386 

3844 
201 

2998 
898 

*  See  Multiplication  Table,  §  39. 


10 


\ 

ARITHMETIC 

7.    62.94 

45.61 

92.36 

*  39.46 

91.64 

32 

46 

63 

68 

25 

8.    3.229 

SU>62 

6.403 

4.899 

1.764 

64 

35 

77 
Exercise  8 

58 

86 

1.  Find  the  cost  of  32  yd.  of  carpet  at  64^  a  yard. 

2.  A  dealer  bought  25  horses  at  an  average  price  of  $75,  and 
48  head  of  cattle  at  $  28  apiece.     Find  the  total  cost. 

3.  A  man  drives  5  mi.  to  the  station,  rides  on  the  train  4  hr.  at 
34  mi.  an  hour,  and  then  drives  into  the  country  6  mi.  What  is 
the  entire  distance  ? 

4.  A  grocer  bought  a  dozen  boxes  of  soap  at  $  2.25  a  box  of 
60  bars,  and  sold  it  for  5  ^  a  bar.     Find  his  gain. 

5.  A  farmer's  wife  bought  4  lb.  of  coffee  at  32  ^  a  pound,  and 
16  lb.  of  sugar  at  5^  ^  a  pound,  giving  in  exchange  8  lb.  of  butter 
at  18  ^  a  pound.  How  much  money  must  she  pay  to  settle  the 
bill? 

6.  How  far  will  a  bicyclist  travel  in  8  da.,  if  he  travels  6  hr.  a 
day  at  9  mi.  an  hour  ? 

7.  Find  the  weight  of  the  hogs  in  a  train  load  consisting  of 
36  cars,  each  car  containing  54  hogs,  the  average  weight  being 
195  lb. 

8.  Find  the  cost  of  a  300-ft.  breakwater  at  $  38  a  foot. 

9.  In  1898  the  State  of  Indiana  employed  237  truant  officers 
at  an  average  salary  of  $169.  Find  the  total  amount  paid  to 
truant  officers. 

10.  A  grain  dealer  bought  6276  bu.  of  wheat  in  Chicago  at 
75  ^  a  bushel,  and  shipped  it  to  New  York  at  a  cost  of  2  ^  a  bushel. 
Find  his  gain  if  he  received  82  ^  a  bushel  for  it  in  New  York. 


DEFINITIONS  AND  REVIEW 


11 


11.  The  city  of  Stockholm  pays  $300,000  a  year  for  regular 
teachers  in  the  public  schools,  and  $300  a  year  to  each  of  304 
substitute  teachers.     Find  the  total  sum  paid. 

12.  It  costs  $30  a  carload  to  remove  watermelons  from  Posey- 
ville,  Ind.,  to  Chicago.  Watermelons  pack  1175  to  the  car ;  how- 
much  less  is  this  per  car  than  3^  a  melon ? 

13.  Find  the  total  weight  of  739  106-lb.  sheep,  954  98-lb.  sheep, 
and  268  62-lb.  lambs. 

14.  Find  the  cost  of  90  260-lb.  hogs  at  $4.64  per  100  lb. 

15.  F'ind  the  value  of  a  carload  of  beeves,  average  weight 
1300  lb.,  at  $  6.40  per  100  lb.,  there  being  19  to  a  carload. 

Exercise  9 

In  the  following  examples  name  the  trial  divisors,  and  find  the 
quotients  and  remainders : 


1. 

21)714 

32)832 

41)3239 

52)4316 

2. 

42)7961 

63)2961 

72)8756 

92)7488 

3. 

19)5344 

38)5192 

83)4472 

69)6365 

4. 

58)8102 

34)2669 

57)7608 

96)2968 

5. 

301)8519 

225)3447 

426)6358 

119)8968 

6. 

108)7400 

213)4966 

125)3608 

524)4853 

7. 

23)66263 

17)15189 

54)52341 

47)72935 

8. 

44)50002 

95)12855 

19)52496 

68)35334 

Exercise  10 

1.  A  bicycle  dealer  bought  6  doz.  bicycles  for  $2016.     Find 
the  cost  of  each  bicycle. 

2.  A  man  paid  $  5025  for  a  farm  of  75  A.     Find  the  cost  per 
acre. 


12  ARITHMETIC 

3.  A  horse  dealer  paid  $  2520  for  72  horses.  What  was  the 
average  price  ? 

4.  If  a  train  travels  34  mi.  an  hour,  how  long  will  it  take  to 
travel  816  mi.,  there  being  stops  amounting  to  45  min.  ? 

5.  The  yield  of  barley  in  a  certain  township  was  32,452  bu., 
the  average  yield  being  28  bu.  to  the  acre.  Find  the  number  of 
acres  of  barley. 

6.  A  golf  course  of  18  holes  is  5886  yd.  around.  Find  the 
average  distance  between  two  holes. 

7.  A  golf  club  spends  $  4000  a  year  to  meet  expenses.  This  is 
met  by  the  annual  dues  of  its  members  at  $  25  apiece.  Find  the 
number  of  members. 

8.  The  total  attendance  at  the  Lincoln  Park  Sanitarium,  Chi- 
cago, for  the  12  weeks  ending  Sept.  9,  1899,  was  125,640.  If  it 
was  open  6  days  a  week,  find  the  average  daily  attendance. 

9.  A  railroad  running  inland  from  the  sea-coast  reaches  in  106 
mi.  an  elevation  of  15,688  ft.  Find  the  average  elevation  in 
feet  per  mile. 

10.  The  following  live  stock  was  received  at  the  Union  Stock- 
yards, Chicago,  in  two  days:  Cattle,  22,097;  hogs,  58,587;  sheep, 
29,659.  Find  the  number  of  carloads  of  each,  if  cattle  average  19 
to  a  car,  hogs  55y  and  sheep  92. 

11.  14,912  men  enlisted  in  the  13  regiments  for  service  in  the 
Philippines.  This  was  1834  more  men  than  were  needed;  find 
the  number  of  men  to  a  regiment. 

12.  A  horse  dealer  sold  32  horses  at  an  average  price  of  $  66 
each,  and  used  the  money  to  buy  24  more.  Find  the  average 
price  paid  for  each  of  the  last  lot. 

13.  A  wholesale  furniture  dealer  sold  144  beds  at  $  16  apiece, 
and  after  using  f  100  of  this  monfey,  bought  sideboards  with  the 
remainder  costing  $  19  each.     How  many  did  he  buy  ? 


DEFINITIONS   AND   REVIEW  13 

14.  A  cattle  dealer  sold  a  herd  of  63  cattle  at  an  average  price 
of  ^  25,  and  with  this  money  bought  12  sheep  at  $  6.25  each,  and 
60  cattle.     Find  the  average  price  of  the  cattle. 

15.  36  carriages  of  the  same  kind  cost  $1332.  What  will 
2  doz.  cost  at  $  5  apiece  more  ? 

16.  An  Indiana  farmer  raised  2294  bu.  of  corn,  averaging  74 
bu.  to  the  acre.  His  neighbor  raised  2294  bu.,  averaging  62  bu.  to 
the  acre.  The  second  farmer  planted  how  many  acres  more  than 
the  first  ? 

17.  A  farmer  received  $439.56  for  the  wheat  he  raised  on 
27  A.,  yielding  22  bu.  per  acre.    Find  the  selling  price  per  bushel. 

18.  An  Ohio  farmer  sold  his  wheat  and  oat  crops  for  1899  for 
$  894.96.  He  raised  625  bu.  of  wheat  and  1854  bu.  of  oats.  If 
the  oats  sold  for  24  /  a  bushel,  find  the  selling  price  of  the  wheat. 


CHAPTER   II 

REVIEW 
Exercise  11 

1.  What  is  J  of  $  12  ?  f  of  §12?  J  of  $15?  J  of  $15?  J  of 
$  24  ?  I  of  $  24  ?     How  do  you  find  f  of  a  quantity  ? 

2.  How  do  you  find  |  of  a  quantity  ?   }?   |?   |?   |? 

3.  What  is  I  of  $  18  ?   |  of  16  oz.  ?   f  of  24  hr.  ?   |  of  24 
sheets  of  paper  ?   |  of  36  in.  ? 

4.  How  many  inches  in  |  ft.  ?     Minutes  in  J  hr.  ?    Quarts  in 
I  bu.  ?     Quarts  in  |  gal.  ?     Inches  in  |  yd.  ? 

5.  A  man  earns  $36  a  week,  and  spends  |  of  it.     What  are 
his  weekly  expenses  ? 

6.  A  farmer  sold  |  of  his  crop  of  624  bu.  of  oats.     How  many 
bushels  did  he  sell  ?     What  did  they  sell  for  at  30  ^  a  bushel  ? 

7.  A  farmer  sold  f  of  his  crop  of  824  bu.  of  corn  at  31  ^  a 
bushel.     What  did  he  receive  for  it  ? 

8.  From  1  yd.  of  ribbon  cut  off  J  yd.,  and  what  part  of  a  yard 
is  left  ?     Cut  off  I  yd.,  and  what  part  is  left  ? 

9.  If  a  man  spends  f  of  his  salary,  what  part  does  he  save  ? 
If  a  boy  sleeps  f  of  a  day,  what  part  of  the  day  is  he  awake  ? 

10.  A  man  earns  $  25  a  week,  and  spends  J  of  it.  What  part 
of  it  does  he  save  ?     How  many  dollars  a  week  ? 

11.  A  lady,  who  had  $24,  spent  f  of  it  for  a  sofa  and  the 
remainder  for  a  chair.  Find  the  cost  of  the  chair.  Give  two 
solutions. 

14 


REVIEW  15 

12.  What  is  i  of  $25.50?   |  of  $25.50?   f  of  $24.60?   |  of 
$19.20?   f  of  $43.40? 

13.  I  paid  $31.50  for  a  bicycle  for  myself,  and  -f  as  much,  for 
a  boy's  bicycle.     Find  the  difference  in  price. 

14.  A  merchant  paid  $  4.32  for  a  rug,  and  sold  it  for  f  of  its 
cost.     Find  how  much  he  lost. 


Exercise  12 

1.  Draw  a  line  1  ft.  long.     Measure  off  a  part  6  in.  long. 
What  part  of  a  foot  is  6  in.  ? 

2.  Make  a  drawing  to  show  that  3  in.  is  i  ft. ;  that  4  in.  is 
i  ft. ;  that  9  in.  is  f  ft. ;  that  8  in.  is  f  ft. ;  that  10  in.  is  f  ft. 

3.  6  is  what  part  of  12?     What  part  of  12  is  4?   8?   9? 
I  is  the  fraction  -^j  reduced  to  its  lowest  terms. 

4.  Reduce  to  their  lowest  terms :  ^^,  t\,  j%,  ||,  ^,  f ,  ^8^, 

6        1  2 
1"JJ    16- 

5.  How  do  you  reduce  a  fraction  to  its  lowest  terms  ? 

6.  Reduce  to  their  lowest  terms :  f,  if,  if,  if,  i|,  f f,  i|,  ||, 

7.  Make  a  drawing  to  show  that  i  of  a  quantity  is  equal  to 
I  of  it. 


8.    Make  drawings  to  show  that : 

4=3  l=-5-  i  — 


2"—  6 


i=A  i=l  \=i 


^•2"  =  ¥                  i  =  iir                   \  —  i  i=iT 

1 ?  1 ?  1 ?  1 ? 

4— TT                          t—  2ir                              "5^  —  TT  t—  2T 

10.    Make  drawings  to  show  that : 

i+i=i              \+i=\  *+i=f 


16  ARITHMETIC 

11.  What  is  the  sum  of : 

iand^?  iaiidi?  iandj?  iandj? 

^andi?  iand^?  Jandi?  iand^? 

^andj?  iandi?  iand|?  iand^? 

12.  Make  drawings  to  show  that  |  =  yV ;  |  =  f ;  z  =  ih 

13.  i  =  i  i  =  TV 

4  _     ?  5  _     ? 

-W  —  TS  T  —  TS 

l  =  T5-  1  =  IT 

14.  Make  drawings  to  show  that : 
(a)   |  +  |  =  1A 

15.  What  is  the  smallest  number  in  the  multiplication  tables 
of  3  and  4  ?  Of  9  and  6  ?  To  what  do  you  reduce  J  and  J  be- 
fore you  can  add  them  ?    ^  and  f  ? 

16.  What  is  the  smallest  number  in  the  multiplication  tables 
of  2  and  5?  3  and  9?  4  and  3?  4  and  6?  5  and  3?  7  and  3? 
4  and  5  ?    9  and  6  ?    7  and  2  ?    8  and  12  ?    3  and  5  ? 

To  what  will  you  reduce  each  fraction  in  the  following  exam- 
ple before  adding  ? 


I—  sV 

f  =  A 

*  +  I  =  tV 

l+l=A 

l+f=iA 

*-*  =  A 

.    Add: 

iand| 

*  and  1 

fandf 

fandf 

1|  and  4f 

iandf 

f  and  1 

3|  and  2f 

4|and2A 

2^  and  3^ 

4J  and  2J 

4§  and  6f 

18.  How  will  you  find  the  difference  between  two  fractions  ? 

19.  Find  the  difference  between : 

fand^       |and^       J  and  ^       fandf 

I  and  J       fandi       f  and  ^      6^  and  3 1 

6}and2f     6J  and  3f     6 1  and  3}     6f  and  4} 


REVIEW  17 

20.  William  had  stamps  worth  $  2^  on  one  penny  savings  card 
and  worth  $  If  on  another.  He  cashed  both  cards,  how  much 
money  did  he  get  ? 

Exercise  13 

1.  A  pitcher  containing  |  gal.  of  water  is  filled  by  pouring 
into  it  ^  gal.  more.     What  part  of  a  gallon  does  it  hold  ? 

2.  Wheat,  bought  at  65^^  per  bushel,  was  sold  at  an  advance 
of  1^^  per  bushel.     Find  the  selling  price. 

3.  Corn,  bought  at  321  /  per  bushel,  sold  for  320.  Find  the 
gain  per  bushel. 

4.  Nov.  24,  1899,  oats  sold  on  the  Chicago  market  for  25^^^ 
per  bushel,  and  the  following  day  the  price  was  f  ^  lower.  Find 
the  price  Nov.  25. 

5.  I  bought  lOf  T.  of  coal  in  the  fall,  and  had  2i  T.  left  over 
the  next  summer.  How  many  tons  did  I  burn  ?  What  did  it 
cost  at  $6  a  ton? 

6.  A  kettle  weighing  3|  lb.  contains  4|-  lb.  of  fruit.  Find 
the  total  weight. 

7.  Two  bicyclists  are  travelling  along  the  same  road,  in  the 
same  direction,  the  first  at  the  rate  of  12^  mi.  an  hour,  the  second 
at  the  rate  of  8|-  mi.  How  many  miles  does  the  faster  gain  on 
the  slower  in  one  hour  ? 

8.  If  turkeys  cost  121^  a  pound,  and  chickens  9i^,  what  is 
the  difference  between  the  cost  of  a  12-lb.  turkey  and  12  lb.  of 
chickens  ? 

9.  A  string  was  cut  into  two  parts,"  one  of  which  is  2|  ft.  long, 
and  the  other  3|.     How  long  was  it  before  being  cut  ? 

10.  I  yd.  of  cloth  costs  60  ^. 

J  yd.  of  cloth  costs  ?  ^. 
I  yd.  of  cloth  costs  ?  ^. 
1  yd.  of  cloth  costs  ?  ^. 

What  is  your  divisor  here  ?     Your  multiplier  ? 


18  ARITHMETIC 

11.  -J  of  my  age  is  21  yr. 
J  of  my  age  is  ?  yr. 

My  age  is  ?  yr. 
What  is  your  divisor  here  ?     Your  multiplier  ? 

12.  What  would  have  been  the  divisor  and  multiplier  in  the 
previous  example  if  the  fraction  had  been  J?    |?    |-?    J?    J? 

13.  I  yd.  of  cloth  costs  30^,  what  is  the  cost  of  1  yd.  ? 

14.  f  of  the  speed  of  a  railway  train  is  12  mi.  an  hour,  find  the 
rate  at  which  it  travels. 

15.  A  man  is  f  as  old  as  his  wife.  If  he  is  48  yr.  of  age, 
what  is  his  wife's  age  ? 

16.  A  boy  said,  "  When  I  am  2|  times  as  old  as  I  am  now,  I 
shall  be  22  yr.  of  age."     How  old  is  he  ? 

17.  A  man  divided  his  farm  between  his  two  sons,  giving  the 
younger  |  of  it.  If  the  older  son  got  90  A.,  what  was  the  size  of 
the  farm  ? 

18.  Harriet  spent  f  of  her  money  for  a  chair,  and  the  re- 
mainder, which  was  $15,  for  a  rug.  How  much  had  she  at 
first? 

19.  Make  examples  like  the  preceding  six  questions,  using  the 
following  price  list : 

Tea  at  60  ^  a  pound.  Eibbon  at  36  ^  a  yard. 

Spoons  at  $  15  a  dozen.  Cloth  at  48  ^  a  yard. 

Exercise  14 

1.  Draw  two  lines,  one  10  in.,  and  the  other  15  in.  long. 
Divide  each  line  into  parts,  each  5  in.  long.  The  ratio  of  10  in. 
to  15  in.  is  |.     The  ratio  of  15  in.  to  10  in.  is  f . 

2.  Make  drawings  to  show  that  the  ratio  of  4  to  6  is  f ;  of 
12  to  i6  is  f ;  of  15  to  18  is  { ;  of  24  to  16  is  f. 


REVIEW  19 

3.  What  is  the  ratio  of  6  to  4  ?  16  to  12  ?  18  to  15  ?  16  to 
24?  18  to  30?  16  to  20?  32  to  36?  28  to  16?  21  to  35? 
33  to  22  ? 

4.  Make  drawings  to  show  that  the  ratio  of  16  in.  to 
1  yd.  1  ft.  is  i ;  of  2  ft.  to  2  ft.  8  in.  is  f . 

5.  What  is  the  ratio  of  8  lb.  to  12  lb.  ?  A  12-lb.  turkey  costs 
$  1.26.  What  part  of  $  1.26  will  an  8-lb.  turkey  cost  at  tlie  same 
price  per  pound  ?     How  much  ? 

6.  What  is  the  ratio  of  9  to  12?  Quails  cost  ^1.40  a  dozen. 
What  part  of  $  1.40  will  9  quails  cost  ?     How  much  ? 

7.  What  is  the  ratio  of  18  to  1  doz.?  At  $5.50  per  dozen, 
what  will  18  geese  cost  ? 

8.  What  is  the  ratio  of  20^  to  18^?  A  grocer  paid  $2.25 
for  eggs  at  18^  per  dozen.  What  will  they  sell  for  at  20^  per 
dozen  ? 

9.  A  grocer  received  $1.12  for  a  jar  of  butter  at  21)^  a  pound. 
What  did  it  cost  him  at  18  ^  a  pound  ? 

10.  I  paid  $4.64  for  Early  Kose  potatoes  at  40^  a  bushel. 
What  would  the  same  quantity  of  Burbanks  have  cost  at  45^ 
a  bushel  ? 

11.  Draw  a  rug  18  in.  by  36  in.  (scale  6  in.  to  1  in.).  Make 
a  drawing  to  show  that  if  this  costs  84  ^,  a  rug  of  the  same 
quality,  36  in.  by  72  in.,  should,  at  the  same  rate,  cost  4  times 
as  much,  or  $  3.36. 

12".  A  Brussels  carpet,  weighing  25  oz.  to  the  yard,  weighs 
altogether  110  lb.  Find  the  weight  of  a  velvet  carpet  of  the  same 
size,  weighing  35  oz.  to  the  yard. 

Exercise  15 

1.  Draw  a  line  .6  of  a  meter  long ;  .4  m.;  .7  m. 

2.  Make  drawings  to  show  lines  .5  m.  long;  .53  m. ;  .67  m. ; 
.89  m. ;  .91  m. 


20  ARITHMETIC 

3.  Make  drawings  to  show  lines  .254  m.  long ;  .875  m. ; 
.624  m. ;  .258  m. 

4.  Make  drawings  to  show  lines  .25  m.  long ;  .33  m. ;  .166  m. 
What  part  of  a  meter  does  each  line  seem  to  be  ? 

5.  3.26  lb.  is  read  three  and  twenty-six  hundredths  pounds; 
5.486  lb.  is  read  five  and  four  hundred  eighty-six  thousandths 
pounds. 

6.  Read  the  following,  supplying  different  units : 

6.42                         2.358  60.006  329.017 

5.84  14.405  412.15  425.7 

3.16  70.4  72.6  19.934 

3.06  70.04  459.245  9.009 

8.2  159.72  300.001  226.043 

24.09                          7.415  721.01  724.981 

16.74                          8.009  455.554  999.999 

7.  In  the  expression  5.429  T.  the  place  value  of  each  figure  is 
given  thus :  5  is  five  tons,  4  is  four  tenths  of  a  ton,  2  is  two  hun- 
dredths of  a  ton,  and  9  is  nine  thousandths  of  a  ton. 

8.  Give  the  place  value  of  each  figure  in  the  following: 
2.5  mi.,  9.34  ft.,  7.258  yd.,  8.04  lb.,  6.104  T.,  64.291  A.,  36.09, 


0.102,  6.056, 

249.003 

• 

Add: 
9.  33.725 

65.876 

274.796 

64.059 

6.932 

989.307 

49.374 

87.04 

891.2 

62.498 

29.985 

56.238 

Subtract : 

10.  3.945 

6.412 

46.279 

62.070 

1.897 

3.547 

38.564 

34.638 

11.   6.29  4.7  8.  846.739 

3.646  3.125  5.273  312.888 


REVIEW 


21 


12.  Find  the  number  of  acres  of  land  in  a  farm  divided  into 
four  fields  containing,  respectively,  14.25  A.,  25.875  A.,  23.75 
A.,  16.125  A. 

13.  From  a  piece  of  cloth  containing  20.125  yd.  were  cut 
12.875  yd.     How  much  was  left  in  the  piece  ? 

14.  From  a  farm  containing  160  A.  the  owner  sold  62.875  A. 
How  much  did  he  still  own  ? 

15.  A  man  bought  a  house  and  lot  with  .375  of  his  money,  and 
invested  the  remainder  in  business.  What  part  of  his  money  was 
invested  in  business  ? 


Multiply : 

1.   2 
3 


Exercise  16 


.32 
3 


.24 
3 


.56 
3 


2.    .213 
4 


.457 
2 


2.47 
6 


5.814 

7 


21.049 
5 


3. 

.7 

6 

.23 

36 

145 

6 

.7 

36 

.23 

.48 

4. 

9.362 

8.364 

54.31 

932 

5.768 

53 

29 

645 

.28 

129 

5.  What  will  15  loads  of   coal  weigh,  the  average  weight 
being  2.375  T.  ? 

6.  A  grocer  sold  16  bbl.  of  apples  a  day  for  one  week  at  a 
gain  of  $  .625  a  barrel.     Find  his  gain  on  a  week's  sales. 

7.  Find  the  cost  of  26  thousand  feet  of  lumber  at  $  34.625 
per  thousand. 

8.  A  merchant  had  $  4875  in  the  bank  and  withdrew  .36  of  it 
to^pay  for  goods.     Find  the  amount  still  remaining  in  the  bank. 

9.  A  drover,  after  selling  part  of  a  herd  of  840  cattle,  had  .55 
of  it  left.     How  many  did  he  sell  ? 


22  ARITHMETIC 

10.  Reduce  to  quarts :  .5  pk.,  .375  pk;,  .875  pk.,  .23  pk.,  .424  pk. 

11.  Find  the  cost  of  .625  pk.  of  beans  at  6^  a  quart. 

12.  Reduce  to  quarts :  .375  bu.,  .75  bu.,  .87  bu.,  .694  bu.,  .825  bu. 

13.  Find  the  cost  of  .875  bu.  of  clover  seed  at  22.5^  a  quart. 

14.  How  many  yards  in  1  mi.  ?  How  many  yards  in  .5  mi.  ? 
.25  mi.  ?     .125  mi.  ?     .35  mi.  ?     .675  mi.  ?     .725  mi.  ? 

15.  How  many  steps  will  a  boy  take  in  walking  .725  mi.,  if  he 
takes  1.5  steps  to  a  yard  ? 

16.  How  many  days  in  .146  of  300  da.  ?  If  there  are  300 
working  days  in  one  year,  how  much  does  a  man  earn  in  .245  yr. 
at  $1.50  a  day? 

17.  How  many  feet  in  1  mi.  ?  In  .125  mi.  ?  What  is  the  cost 
of  laying  a  concrete  sidewalk  .125  mi.  long  at  $  .625  a  foot  ? 

18.  How  many  quarts  in  .375  bu.  ?  What  is  the  cost  of  .375 
bu.  of  flaxseed  at  4.75^  a  quart  ? 

Exercise  17 

1.  Divide  6.969  by  3 ;  4.864  by  2 ;  9.356  by  4. 

2.  Find  the  value  of : 

4.864-4  9.465-5  19.35-2 

39.588-6  8.0     -5  231.      -^4 

27.291-3  37.        -4  2150.42-4 

.994  -  7  2.520-8  721 .44  -  3 

2.097-9  54.76    -8  59.2   -8 

In  the  first  two  divisions  what  is  the  place  value  of  each  figure 

in  t^'^  quotient? 

3.  A  vessel  travels  54.5  knots  in  4  hr.,  what  is  the  rate  per 
hour  ? 

4.  1  gal.  contains  231  cu.  in.,  how  many  cubic  inches  are  there 
in  1  qt.  liquid  measure  ? 

6.    1  pk.  contains  537.6  cu.  in.,  how  many  cubic  inches  in  1  qt. 
dry  measure  ? 


REVIEW 


23 


6.  A  quart  of  oats  occupies  how  many  more  cubic  inches  than 
a  quart  of  water  ? 

7.  Find  the  value  of  : 


178.92    - 

-21 

212.16    - 

-68 

70.578  - 

-27 

9.798  - 

-46 

13.088  - 

-32 

362.79 


87 


132.634  --  94 

876.6      ^  18 

4.15    ^25 

18.468  ~  76 

8.  A  farmer  got  18  loads  of  hay  from  one  field.     If  the  total 
amount  was  15.57  T.,  find  the  average  weight  of  one  load. 

9.  A  farmer  sold  36  bu.  of  oats  for  $8.46.    Find  the  price  per 
bushel. 

10.    Show  by  measuring  that  12  m.  divided  by  4  m.  gives  the 
same  quotient  as  1.2  m.  divided  by  .4  m.     Why  is  this  so  ? 


11 

18  in.  --  3  in.  =  ? 

1.8  in.  --  .3  in.  =  ? 

24  yd.  --  4  yd.  =  ? 

2.4  yd.  -h  .4  yd.  =  ? 

1.5  -  .3  =  ? 

9.6  -  .8  =  ?                    4.8  -  .6  =  ? 

4.9  -  .7  =  ? 

7.2 -.9  =  ?                    3.5 -.5  =  ? 

12 

Divide  4.548  by  .6. 

.6 

4.o48            Move  the  decimal  point  one  place  to  the  ri^ht  in  both 

6 

45.48       divisor  and  ( 

iividend,   and  divide  45.48  by  6,  as  in  the 

13.    Divide  157.324  by  .74. 
.74)  157.324  ( 
74)15732.4(212.6 


148 


93 

74  ' 

192 

148 
444 
444 


Move  the  decimal  point  two  places  to  the  right 
in  both  divisor  and  dividend  before  dividing.  Why 
can  you  do  this  ? 


24 


ARITHMETIC 

14.   Find  the  value  of 

: 

7.932  --  .6 

10.101  -f-  3.9 

16 -.8 

4.536  ^  .12 

149.85  ^  .37 

36  ^  .09 

43.05  ^  .15 

.936  ^  .15 

6 -5- .75 

12.186  -  .18 

5963.6  --  6.8 

36  -i-  .45 

1.75  ^  .05 

75.250  -  86 

150  --  .25 

3.78  -  6 

59.328  --  .48 

27  -^  .09 

15.  A  man  divided  a  field  containing  4  A.  into  lots,  each  con- 
taining .25  A.     Find  the  number  of  lots. 

16.  Eeduce  to  yards :  12  ft.,  63  ft.,  6.3  ft.,  25.2  ft.,  8.58  ft, 
.741  ft. 

17.  Eeduce  to  feet :  48  in.,  624  in.,  50.4  in.,  3.936  in.,  3.876  in., 
9.168  in. 

Exercise  18 

1.  Draw  a  line  8  in.  long,  and  divide  it  into  fourths.     Count 
the  number  of  units  in  the  line.     How  many  ? 

2.  A  quantity  considered  in  percentage  is  measured  by  how 
many  units  ? 

3.  What  is  1  of  100  ?     ^  of  100?     f  of  100  ?     J  of  100? 

4.  1^  of  a  quantity  equals  what  per  cent  of  it?     ^?     |?     ^? 

5.  What  part  of  a  quantity  is  25  %  of  it  ?     50  %  ?     75  %  ? 
100%  ? 

6.  Find   50%    of   f  48;  25%   of  $36;  75%   of  $36;  100% 
of  1 15. 

7.  A  farmer  gave  his  son  25%  of  his  farm  of  640  A.     What 
was  the  size  of  his  son's  farm  ? 

8.  A  grocer  paid  28^  a  lb.  for  coffee,  and  sold  it  so  as  to  gain 
25  %  ;  find  the  gain.     Find  the  selling  price. 

9.  My  furnace  burns  12  T.  of  coal  during  the  winter  and  my 
neighbor's  75%  as  much.     How  much  coal  does  he  burn  ? 


REVIEW  25 

10.  A  merchant  paid  32^  a  yard  for  cloth  and  sold  it  at  a 
gain  of  8^  a  yard.  What  part  of  the  cost  did  he  gain?  What 
per  cent  ? 

11.  25  %  of  a  man's  age  is  12  yr.,  how  old  is  he  ? 

12.  What  is  i  of  100  ?  |  of  100  ?  J  of  a  quantity  is  what 
per  cent  of  it?     f?     J?    f?     |? 

13.  A  man  built  a  house  which  cost  $  2250  on  a  lot  that  cost 
66|%  of  that  sum.  Find  the  cost  of  the  lot.  Find  the  cost  of 
the  house  and  lot  together. 

14.  A  grocer  bought  peaches  at  90^  a  bushel,  and  sold  them  at 
a  gain  of  33|^  %.     Find  the  selling  price  per  bushel.     Per  peck. 

15.  What  part  of  $  36  is  $  12  ?  A  merchant  bought  rugs  at 
$  36  each,  and  sold  them  at  an  advance  of  $  12.  Find  his  gain 
per  cent. 

16.  A  speculator  gained  $  400  on  a  lot  by  selling  it  at  a  gain 
of  66|  %.     Find  what  he  paid  for  it. 

Exercise  19 

1.  What  is  i  of  100?  |  of  100  ?  f  of  100?  f  of  100? 
I  of  100  ? 

2.  Commit  to  memory  :  20%  =i  40%  =f,  60%=f,  80%=f 
What  is  meant  by  saying  that  20%  =  J  ? 

3.  What  is  i  of  100  ?  f  of  100  ?  |  of  100  ?  |  of  100  ? 
i  of  100  ? 

4.  Commit  to  memory:  i  =  12i%,  f  =  37i%,  |  =  62i%, 
1  =  871%,  i  =  16|%. 

5.  How  many  minutes  in  20%  of  1  hr.  ?  Quarts  in  12i%  of 
1  bu.  ?  Ounces  in  621  %  of  1  lb.  avoir.  ?  Hours  in  37^  %  of 
1  da.  ?     Square  inches  in  16|%  of  1  sq.  ft.  ? 

6.  yV  of  100%  =  ?  JgOflOO%=^  Jg  of  100%  =  ?  Mem- 
orize: TV  =  8i%,  rV==6i%,TV  =  6|%. 


26  ARlTHiMETIC 

7.  What  part  is  3  in.  of  1  yd.  ?  What  per  cent  ?  What  per 
cent  is  $5  of  ^20? 

8.  A  man  paid  $640  for  a  lot,  and  sold  it  at  a  gain  of  6\%. 
Find  the  gain  and  the  selling  price. 

9.  A  man  sold  a  cow  for  j  of  the  cost  price.  Find  his  loss 
per  cent. 

10.  What  per  cent  is  gained  by  selling  goods  at  |  of  the  cost  ? 
At  I  of  the  cost  ?     At  f  of  the  cost  ? 

11.  Cloth  which  cost  30^  a  yard  was  sold  at  a  loss  of  16|%. 
Find  the  selling  price. 

12.  A  merchant  paid  60^  a  yard  for  cloth,  and  sold  it  at  75^ 
a  yard.     Find  his  gain.     Find  his  gain  per  cent. 

13.  What  per  cent  is  gained  if  cloth  costing  40^  a  yard  is  sold 
for48^ayard?     50^?     44^?     60^?     45^?     55^? 

14.  What  per  cent  is  lost  if  cloth  costing  48^  a  yard  is  sold  for 
36^  a  yard?     40^?     42^?     30^? 

Miscellaneous  Exercise  20 

1.  The  number  of  cars  of  grain  received  in  Chicago,  Sept.  8, 
1899,  was  as  follows :  winter  wheat,  37 ;  spring  wheat,  54 ;  corn, 
690;  oats,  323;  rye,  9;  barley,  40.     Find  the  total  number. 

2.  During  the  year  1899  the  different  railroads  entering 
Chicago  elevated,  respectively,  2  mi.,  2  mi.,  21  mi.,  12  mi.,  12  mi., 
1  mi.,  11  mi.,  8  mi.,  8  mi.,  and  4  mi.,  of  track.  Find  the  total 
number  of  miles  of  track  elevated  in  1899. 

3.  Find  the  amount  of  the  following  bill : 

12  yd.  cotton  at  6^  ^  a  yard ; 
8  yd.  lining  at  11^  a  yard; 
6  yd.  dimity  at  22  ^  a  yard ; 
1  doz.  towels  at  25  ^  each. 


REVIEW  27 

4.  A  speculator  bought  28  A.  of  land  at  $54  an  acre,  and 
35  A.  at  $  65  an  acre.  If  he  sold  it  all  at  $  63  an  acre,  find  his 
gain. 

5.  If  25  cars  of  spring  wheat  contain  51,000  bu.,  find  the 
average  number  of  bushels  to  the  car. 

6.  If  116  cars  "of  corn  contain  278,400  bu.,  find  the  average 
number  of  bushels  to  the  car. 

7.  In  the  city  of  Chicago  in  1899  there  were  243.57  mi.  of 
sidewalk  built  and  67.88  mi.  repaired.  Find  the  total  number  of 
miles  built  and  repaired. 

8.  Make  a  drawing  to  show  that  |  ft.  -+- 1  ft.  =  1 J  ft. 

9.  Two  boys  picked  the  fruit  oif  a  cherry  tree.  One  picked 
1  bu.  and  the  other  f  bu.     How  much  fruit  did  the  tree  bear  ? 

10.  What  part  is  $2  of  $5?     What  per  cent?     What  per 
cent  is  f  1  of  $3?     $2  of  $3?     $3  of  $4? 

11.  A  drover  bought  sheep  at  3  5  each,  and  sold  them  at  an 
advance  of  $  1.     Find  his  gain  per  cent. 

12.  A  fruit  dealer  bought  oranges  at  24^  a  dozen,  and  sold 
them  for  3  ^  each.     Find  his  gain  per  cent. 


CHAPTER   III 

NUMERATION  AND   NOTATION 

4.  Numeration  is  counting,  or  the  expression  of  number 
in  words. 

The  ordinary  system  of  numeration  is  the  Decimal  System^ 
so  called  because  it  is  based  on  the  number  ten. 

5.  The  names  of  the  first  group  of  numbers  in  regular 
succession  are :  one,  two,  three,  four,  five,  six,  seven,  eight, 
nine. 

Other  number  names  are  :  ten,  hundred,  thousand,  million, 
billion,  trillion,  etc. 

6.  The  number  one  applied  to  any  unit  denotes  a  quantity 
which  consists  of  a  single  unit  of  the  kind  named. 

The  number  next  following  nine  is  ten,  which  applied  to 
any  unit  denotes  a  quantity  consisting  of  nine  such  units 
and  one  unit  more. 

The  number  hundred  applied  to  any  unit  denotes  a  quan- 
tity which  consists  of  ten  ten-units. 

The  number  thousand  applied  to  any  unit  denotes  a  quan- 
tity which  consists  of  ten  hundred-units. 

The  number  million  applied  to  any  unit  denotes  a  quantity 
which  consists  of  a  thousand  thousand-units. 

7.  The  number  tenth  applied  to  any  unit  denotes  that 
quantity  of  which  ten  make  up  the  unit. 

The  number  hundredth  applied  to  any  unit  denotes  that 
quantity  of  which  one  hundred  make  up  the  unit. 


NUMERATION  AND  NOTATION  29 

The  number  thousandth  applied  to  any  unit  denotes  that 
quantity  of  which  one  thousand  make  up  the  unit. 

8.  Instead  of  speaking  of  "  the  number  denoted  by  "  5,  75, 
or  375,  we  may  for  brevity  speak  of  the  number  5,  75,  or  375. 

9.  Notation  is  the  art  of  expressing  numbers  by  means  of 
certain  number  symbols  called  numerals  or  figures. 

10.   The  Arabic  Numerals,  styled  also  Figures,  are 

0,      1,      2,      3,      4,      5,      6,      7,      8,      9, 

denoting  naught,  one,  two,  three,  four,  five,  six,  seven,  eight,  nine 
respectively.  The  first  of  these  is  called  naught,  cipher,  or 
zero;  the  remaining  nine  are  called  digits.  By  means  of 
these  numerals  and  a  dot  called  the  decimal  point,  we  can 
write  down  any  number  expressed  decimally.  The  method 
of  doing  so  may  be  described  as  follows : 

A  figure  immediately  to  the  left  of  the  decimal  point 
denotes  so  many  single  units. 

A  figure  immediately  to  the  left  of  the  single-units  figure 
denotes  so  many  tens  of  the  units,  while  a  figure  immediately 
to  the  right  of  the  single-units  figure  denotes  so  many  tenths 
of  the  unit. 

Figures  to  the  left  of  the  tens-figure,  taking  them  in  order 
from  right  to  left,  denote  so  many  hundreds  of  the  unit,  so 
many  thousands  of  the  unit,  so  many  ten-thousands  of  the 
unit,  etc. 

Figures  to  the  right  of  the  tenths-figure,  taking  them  in 
order  from  left  to  right,  denote  so  many  hundredths  of  the 
unit,  so  many  thousandths  of  the  unit,  so  many  ten-thousandths 
of  the  unit,  etc. 

The  function  of  the  decimal  point  is  to  mark  the  place  of 
the  standard  unit  when  the  quantity  is  measured. 


80 


11. 


ARITHMETIC 

following  qi 

lantit 

9 

yd. 

69 

(4 

259 

U 

3259 

ii 

43259 

u 

843259 

u 

59.7 

(( 

59.76 

(( 

59.761 

it, 

59.7613 

ii 

In  the  above  quantities 

9  denotes  9  of  the  unit  one  yard. 


5 

5 

2 

2 

3 

3 

4 

4 

8 

8 

7 

7 

6 

6 

1 

1 

3 

3 

ten  yards. 

one  hundred  yards. 

one  thousand  yards. 

ten  thousand  yards. 

one  hundred  thousand  yards. 

one  tenth  of  a  yard. 

one  one-hundredth  of  a  yard. 

one  one-thousandth  of  a  yard, 

one  ten-thousandth  of  a  yard. 


12.  The  number  8  always  denotes  8  of  the  unit ;  86  de- 
notes 8  of  the  ten-unit  or  80  of  the  unit  and  6  of  the  unit, 
and  is  read  eighty-six  of  the  unit. 

865  denotes  8  of  the  hundred-unit  and  6  of  the  ft'??-unit 
and  5  of  the  unit;  i.e.  800,  60,  and  5  of  the  unit,  and  is  it  ad 
eight  hundred  sixty-five  of  the  unit. 

Thus,  the  numbers  8,  86,  865  always  denote  eight,  eiglity- 
six,  eight  hundred  sixty-five  respectively,  the  position  of  the 
figures  in  each  case  giving  the  unit.     For  example  : 


NUMERATION  AND  NOTATION  31 

In  the  number  865,865,865  each  865  is  read  eight  hundred 
sixty-five,  the  difference  being  in  the  unit  only ;  865  of 
tlie  million-unit,  865  of  the  thousand-unit,  and  865  of  the 
one-unit. 

This  number  is  read  eight  hundred  sixty-five  million  eight 
hundred  sixty-five  thousand  eight  hundred  sixty-five. 

Exercise  21 

1.  Express  in  words  the  numbers  given  in  Exercises  28,  30, 
38,  39,  41. 

Read  the  following  statements : 

2.  During  the  year  1899  about  6,349,662,320  pieces  of  mail 
matter  were  posted  in  the  United  States.  Of  this  number  6,312,- 
732  were  sent  to  the  dead  letter  office. 

3.  To  show  the  increase  in  the  paper-bag  business  it  is  said 
that  the  sales  for  the  United  States  were  in  1871  about  391,000,000 
a  year,  and  in  1897  the  sales  were  3,943,000,000.  The  sale  for 
the  year  1899  was  about  5,000,000,000. 

4.  During  the  ten  months  ending  Oct.  31,  1899,  the  exports 
from  the  United  States  were  valued  at  $  1,029,242,286,  and  beat 
the  record  by  $  41,344,597. 

5.  For  the  year  ending  Oct.  31, 1899,  the  exports  were  $  1,296,- 
890,945,  an  increase  of  more  than  $  70,000,000  over  the  previous 
year. 

Exercise  22 

Express  in  words : 

1.  .7  ton;  .64  ton  ;  .643  ton.        3.    9.403  hr. ;  29.04  min. ;  .09  sec. 

2.  4.92  lb. ;  8.09  lb. ;  2.734  lb.    4.    7.456  A. ;  6.7985  A. 

5.  8452.69  sq.  mi. ;  21.4394  A. 

6.  Express  in  words  the  numbers  given  in  Exercises  111  and  112. 


32  ARITHMETIC 

Exercise  23 
Write  in  figures : 

1.  Three  hundred  forty-nine;  eight  thoiisand  four  hundred 
sixty-nine;  nine  thousand  five  hundred  seventy. 

2.  Twenty-nine  thousand  one  hundred  thirty-four;  fifty  thou- 
sand eight  hundred  seventy-six;  seventy-eight  thousand  three 
hundred. 

3.  Nine  hundred  fifty-two  thousand  seven  hundred  forty  ;  six 
hundred  forty-nine  thousand  nine  hundred  five;  nine  hundred 
thousand  eight  hundred  sixty-four. 

4.  One  hundred  sixty-eight  thousand  six  hundred  eighteen; 
three  hundred  twelve  thousand  seven  hundred  forty-two;  four 
hundred  sixty-one  thousand  eight  hundred  twenty-one. 

5.  Seven  hundred  fourteen  thousand  thirty;  three  hundred 
thousand  two  hundred  four ;  one  hundred  thousand  fifty. 

6.  Eight  hundred  ninety  thousand  one  hundred  twenty-three ; 
two  hundred  four  thousand  six  hundred  seventy-eight ;  nine  hun- 
dred one  thousand  two  hundred  thirty. 

7.  Sixteen  thousand  sixty ;  seven  hundred  two  thousand  nine 
hundred  five;  one  hundred  seventy-five  thousand  two  hundred 
fifteen. 

8.  Ninety  thousand  twenty-three;  five  hundred  twenty  thou- 
sand sixty-four ;  three  hundred  twenty-four  thousand  four  hundred 
forty-four. 

Exercise  24 
Express  in  figures : 

1.  Five  tenths;  eight  tenths;  one  tenth;  six  and  four  tenths; 
twenty-five  and  nine  tenths ;  twenty  and  two  tenths. 

2.  Forty-five  hundredths;  sixty-seven  hundredths;  twenty- 
seven  hundredths ;  seven  hundredths ;  nine  hundredths ;  two 
hundredths;   six  and  thirty-four  hundredths. 


THE  ROMAN  NOTATION  33 

3.  Six  and  three  tenths;  sixty-three  hundredths;  ninety-five 
hundredths ;  nine  and  five  tenths ;  two  and  four  tenths ;  two 
and  four  hundredths  ;  eight  tenths  ;  eight  hundredths. 

4.  Six  hundred  twenty -five  thousandths;  five  hundred  eight 
thousandths ;  two  hundred  forty-seven  and  eight  hundred  six 
thousandths  ;  nine  and  fifteen  thousandths ;  six  thousandths. 

6.  Five  tenths;  two  and  sixty-seven  hundredths;  nine  hun- 
dredths ;  fourteen  thousandths  ;  sixteen  thousandths. 

6.  Three  and  twelve  thousandths ;  three  thousand  four  hundred 
fifty-three;  six  thousand  eight  hundred  seventy-five  ten-thou- 
sandths ;  two  and  one  hundred  ninety-nine  ten-thousandths. 

7.  Nine  hundred  six  thousandths ;  nine  hundred  six  ten-thou- 
sandths ;  twenty  and  forty-five  thousandths  ;  seventeen  and  seven 
ten-thousandths;  three  hundred  three  and  nine  ten- thousandths. 

THE   ROMAN   NOTATION 

13.  The  Arabic  Notation  is  the  one  in  general  use.  It 
wJls  introduced  into  Europe  by  the  Arabs.  The  system  of 
notation  which  was  used  among  the  Romans  is  now  used 
only  to  denote  the  chapters  and  sections  of  books,  etc. 

14.  The  following  letters  are  used  to  denote  numbers,  and 
their  values  are  written  below  : 


I 

V 

• 

X 

L 

C 

D 

M 

1 

5 

10 

50 

100 

500 

1000 

15.   The  numbers  6,  8,  15,  20  are  represented  thus  :    ■ 

VI        VIII       XV        XX 

Hence  if  a  character  in  the  Roman  Notation  be  followed 
by  another  of  equal  or  less  value,  the  number  denoted  by 
the  expression  is  equal  to  the  sum  of  the  simple  values. 


84  ARITHMETIC 

16.  The  numbers  4,  9,  40,  and  90  are  represented  by  IV, 
IX,  XL,  XC. 

Hence  if  a  character  in  the  Roman  Notation  is  followed 
by  one  of  greater  value  than  itself,  the  number  denoted  by 
the  expression  is  the  difference  of  their  simple  values. 

17.  Express  1896  in  Roman  numerals. 

1896  =  1000,  800,  90,  and  6 
1000  =  M 
800  =  DCCC 
90  =  XC 
6  =  VI 
.-.  1896  =  MDCCCXCVI 

Hence  to  write  any  number  in  Roman  numerals,  separate 
the  number  into  its  different  parts,  and  write  down  the  parts 
in  order,  beginning  at  the  left. 

Exercise  25 
Write  in  Roman  numerals : 

1.  14,  25,  54,  89,  99. 

2.  178,  304,  871,  982,  999. 

3.  1204,  1590,  1756,  1876,  1895. 

Write  in  figures : 

4.  XLVI,  LXXIX,  XCIV,  LXXXIII. 

5.  XCIX,  CXXXIX,  CLX. 

6.  DLIV,  MDCII,  MDCCCXIX,  MXC. 


CHAPTER   IV 

ADDITION 

18.  Let  the  length,  of  a  room  be  measured  by  the  parts, 
2  ft.,  3  ft.,  4  ft.,  and  5  ft.  Here  the  common  unit  of  meas- 
ure, 1  ft.,  has  been  repeated  2,  3,  4,  and  5  times  to  measure 
the  parts. 

The  number  of  units  in  all  is  the  sum  found  primarily  by 
counting  2,  3,  4,  and  5,  or  14  units  of  1  ft.  Hence  the  length 
of  the  room  which  is  now  definitely  measured  is  14  ft. 

Addition  may,  therefore,  be  considered  as  the  operation  of 
finding  the  quantity,  which,  as  a  whole,  is  made  up  of  two  or 
more  given  quantities  as  its  parts.  Each  of  these  quantities 
must  have  the  same  measuring  unit.  Not  only  is  it  im- 
possible to  add  5  ft.  to  4  min.,  it  is  impossible  to  add  5  ft. 
to  4  in.;  i.e.  to  express  without  change  of  unit  the  whole 
quantity  by  a  number  of  either  feet  or  inches. 

The  patts  added  are  called  Addends. 

The  Sum  is  the  quantity  obtained  by  adding  the  quantities 
expressed  in  terms  of  a  common  unit. 

19.  The  Sign  of  Addition  is  +,  and  is  read  plus;  thus 
6  4-  8  is  read  6  plus  8. 

The  Sign  of  Equality  is  =,  and  is  read  equals  or  equal; 
thus  4  +  5  =  9  is  read  4  plus  5  equals  9. 

20.  I  bought  3  farms  of  50  A.  each,  6  farms  of  50  A., 
and  4  farms  of  50  A.     How  much  did  I  buy  altogether  ? 

Here  we  are  required  to  find  the  whole  quantity  measured  by  the  sum  of 
3,  6,  and  4  farms  of  50  A. 

.*.  the  whole  quantity  =  13  farms  of  60  A. 

35 


36  ARITHMETIC 

Exercise  26 

1.  What  quantity  is  measured  by  the  parts  2  in.,  5  in.,  and 
4  in.? 

2.  3  ten-dollar  bills  +  4  ten-dollar  bills  -f  6  ten-dollar  bills  =  ? 
ten-dollar  bills.     How  many  dollars  ? 

3.  How  many  five-cent  pieces  are  equal  to  4  five-cent  pieces, 
9  five-cent  pieces,  and  5  five-cent  pieces  ?    How  many  cents  ? 

4.  What  is  the  quantity  denoted  by  the  sum  6,  7,  and  5  times 
the  measuring  unit  ? 

5.  I  paid  out  in  one  day  6  ten-dollar  bills,  8  ten-dollar  bills, 
and  5  ten-dollar  bills.     How  much  did  I  spend  all  together  ? 

6.  If  I  sell  two  lots,  one  for  8  units  of  value,  and  the  other 
for  6  units,  what  do  I  get  for  both,  the  unit  of  value  being  $  100  ? 

7.  A  fruit  dealer  who  arranges  his  apples  in  piles  of  6  for  5^, 
sells  1  pile  to  each  of  a  company  of  4  persons,  and  3  piles  to 
another  customer.  How  much  does  he  sell  all  together?  How 
many  apples? 

8.  A  speculator  bought  5  farms  of  100  A.  for  $  5000,  6  farms 
of  100  A.  for  ^  7000,  and  3  farms  of  100  A.  for  ^  4000.  How 
much  land  did  he  buy  ?    What  did  it  cost  ? 

9.  A  fruit  dealer  sells  his  apples  at  the  rate  of  3  for  5  cents. 
He  sells  5  cents'  worth  to  each  of  8  customers.  How  many  apples 
did  he  sell  ? 

10.  1  gal.  3  qt.  =  ?  qt.      3  gal.  1  qt.  =  ?  qt.     What  is  the  cost 
of  2  gal.  3  qt.  of  milk  at  6^  a  quart  ? 

11.  1  pk.  6  qt.  =  ?  qt.     4  pk.  2  qt.  =  ?  qt.    What  is  the  cost  of 

1  pk.  4  qt.  of  berries  at  7^  a  quart  ? 

12.  1  yd.  2  ft.  =  ?  ft.     3  yd.  2  ft.  =  ?  ft.     Find  the  weight  of 

2  yd.  2  ft.  of  carpet  if  1  ft.  weighs  7  oz. 

13.  How  many  working  days  in  1  wk.  5  da.  ?    What  will  a 
man  earn  in  1  wk.  4  da.  at  f  1.50  a  day  ? 


ADDITION  37 

14.  1  bu.  8  qt.  =  ?  qt.     1  bu.  12  qt.  =  ?  qt.     I  bought  1  bu. 
6  qt.  of  cherries  at  5  ^  a  quart      Find  the  cost. 

15.  1  doz.  +  3  =  ?     2  doz.  +  6  =  ?     What  did  I  pay  for  1  doz. 
and  6  eggs  at  2  ^  apiece  ? 

21.   Drill  on  the  following  addition  combinations  to  secure 
accuracy  and  rapidity : 

1112    12    123    123    1234 
1232    43    543    654    7654 


1    2 

8    7 

3    4          1 
6    5         9 

2    3    4    5 

8    7    6    5 

2    3    4    5 

9    8    7    6 

3    4    5    6 

9    8    7    6 

4    5 

9    8 

6  5    6 

7  9    8 

7          6    7 
7          9    8 

7    8          8 
9    8          9 

9 
9 

Enlarge  each  combination  thus : 

8  8      8  •       8    18    28  18    38    68 

9  19    29,  etc.  ^    _9    _9^ etc.  19    29    39,  etc. 

80      80  80    180    280 

90    190,  etc.  90     _90     _90,  etc. 

Give  such  problems  as  the  following,  requiring  instan- 
taneous answers : 

How  many  square  feet  in  1  sq.  yd.  8  sq.  ft.  ?  1  sq.  yd. 
6  sq.  ft.  ?  etc. 

How  many  quarts  in  1  pk.  4  qt.  ?  etc. 

When  7  is  the  number  added,  base  the  problem  on  days, 
thus  : 

How  many  days  in  1  wk.  4  da.  ? 

When  the  number  is  6,  on  minutes,  thus : 

How  many  seconds  in  1  min.  30  sec? and  so  on. 


88  ARITHMETIC 

Exercise  27 
Count  by : 

1.  2's  from  0  to  100 ;  1  to  101. 

2.  3's  from  0  to  102 ;  1  to  103 :  2  to  104. 

3.  4's  from  0  to  100 ;  1  to  101 ;  2  to  102 ;  3  to  104. 

4.  5's  from  0  to  100 ;  1  to  101 ;  2  to  102 ;  3  to  103;  4  to  104. 

5.  6's  from  0  to  102 ;   1  to  103 ;  2  to  104 ;  3  to  105 ;  4  to  106 ; 
5  to  107. 

6.  7's  from  0  to  105  j  1  to  106 ;    2  to  107 ;  3  to  108 ;  4  to  109. 

7.  8's  from  0  to  104 ;  1  to  105 ;    2  to  106  ;  3  to  107 ;  4  to  108. 

8.  9's  from  0  to  99;   1  to  100;    2  to  101;   3  to  102;   4  to  103. 

22.   A  person  paid  $  38  for  a  cow,  1 146  for  a  horse,  and 
f  255  for  a  carriage.     Find  the  cost  of  all. 

^    38  In  this  problem  we  are  required  to  find  the  cost  which  is  the 

146  "'^^ole  measured  by  the  parts  $  38,  f$  146,  and  $  255. 

_^  This  may,  for  convenience,  be  broken  up  into  the  sum  of  5,  6, 

^^^  and  8  units  of  $  1,  5,  4,  and  8  units  of  $  10,  and  2  and  1  units  of 


$439  $  1^^-  ^^^  ^^^  ^^  ^'  ^'  ^^^^  ^  ^^^^^^  of  $  1  =  19  units  of  $  1  =  1 
unit  of  $  10  and  9  of  the  $  1  unit. 

Add  the  1  unit  of  $  10  in  with  the  tens'  column. 

The  sum  of  1,  5,  4,  and  3  units  of  §  10  =  13  units  of  $  10,  or  1  unit  of 
$  100,  and  3  units  of  $  10. 

Add  the  1  unit  of  $  100  in  with  the  hundreds'  column. 

The  sum  of  1,  2,  and  1  unit  of  $  100  =  4  units  of  $  100. 

Hence  the  cost  =  4  units  of  $  100,  3  units  of  $  10,  and  9  units  of  $  1  =  $439. 

23.  Find  the  sum  of  the  following  numbers,  using  the  unit 
employed  in  stating  your  age : 

234'  .'.  the  sum  =  2626  yr.,  since  1  yr.  is  the  unit  of  age. 

qoo  Add  thus,  beginning  at  the  units'  column  :  0,  12,  16  ;  writedown 

6  and  carry  1  to  the  tens'  column;  5,  11,  19,  22  ;  write  down  2 
^^ '  under  the  tens'  column  and  carry  2  to  the  hundreds'  column ;  10,  15, 
842  24,  26  ;  write  down  26,  putting  the  6  under  the  hundreds'  column. 
To  prove  the  answer  correct,  add  downward.      If  the  same 

answer  is  obtained,  the  result  is  likely  to  be  correct. 


2626 


ADDITION 


39 


Exercise  28 

Find  the  sum  of  the  following  quantities  and  explain  your 
work  clearly.  Prove  each  answer  correct  by  beginning  at  the  top 
and  adding  down. 

2.    341  ct.  3.    532  4.    687 

225  "  233  956 

343  "  154  888 

5. 


9. 


13. 


17. 


543  min. 

6.  635  hr. 

7. 

247 

8. 

976 

666  " 

87  " 

859 

485 

752  " 

256  " 

23 

329- 

231  " 

742  « 

271 

786 

2576 

10.  2598 

11. 

4397 

12. 

8649 

3491 

6776 

8999 

6262 

7743 

4259 

5637 

8497 

8988 

7362 

8249 

9773 

64251 

14.  89435 

15. 

850439 

16. 

262933 

3789 

62789 

973642 

998757 

45278 

576 

845867 

639364 

99 

43 

939894 

286753 

6472 

416 

768795 

486325 

2573 

7235 

649879 

744638 

634879 

18. 

453'/!98 

19. 

296843 

273548 

667788 

796276 

506644 

549763 

947362 

979721 

438925 

288433 

346294 

648888 

334455 

434696 

999999 

667854 

20.  How  many  days  in  1  leap  year  ? 

Find  the  number  of  days  in  1  leap  year  and  257  da. 

21.  How  many  square  inches  in  1  sq.  ft.  ? 

Find  the  number  of  square  inches  in  1  sq.  ft.  96  sq.  in. 
Find  the  number  of  square  feet  in  1  sq.  yd.  6  sq.  ft. 


40  ARITHMETIC 

22.  How  many  yards  in  1  mi.  ? 

How  many  steps  will  a  man  take  in  walking  1  mi.  468  yd.,  if 
he  goes  1  yd.  in  each  step  ? 

23.  How  many  feet  in  1  mi  ? 

How  many  feet  apart  are  two  places  the  distance  between  which 
isl  mi.  3480  ft.? 


Add: 

• 

24.   84 

25.  99 

26.   893 

27.   733 

93 

77 

254 

842 

89 

86 

767 

951 

75 

25 

899 

258 

91 

43 

654 

365 

27 

88 

473 

874 

30 

76 

129 

935 

45 

52 

895 

273 

28.  542 

29.  9834 

30.  7594 

31.  5846 

879 

729 

821 

7593 

666 

8345 

2357 

3819 

257 

728 

8463 

5578 

389 

3403 

1525 

2904 

983 

17 

7469 

8392 

365 

295 

2856 

9576 

874 

8943 

8888 

2882 

24.   1  mi.  contains  5280  ft. ;  find,  by  adding,  the  number  of 
feet  in  4  mi. 

Here  we  are  required  to  find  the  sum  of  four  equal  addends,  each  of  which 
is  5280  ft. ;  thus : 

5280  ft. 
5280  ft. 
5280  ft. 
5280  ft. 
21120  ft. 

Multiply  6280  ft.  by  4  and  prove  this  answer  correct. 


ADDITION  41 

Exercise  29 

1.  1  bu.  contains  32  qt.,  find,  by  adding,  the  number  of  quarts 
in  4  bu.     Multiply  32  qt.  by  4  and  prove  your  answer  correct. 

2.  1  mi.  contains  1760  yd.,  find,  by  adding,. the  number  of  yards 
in  3  mi.     Multiply  and  prove  your  answer  correct. 

3.  One  square  foot  contains  144  sq.  in.  Find,  by  adding,  the 
number  of  square  inches  in  a  rug  containing  6  sq.  ft.  Multiply 
and  prove  your  answer  correct. 

4.  How  many  hours  in  one  day  ?  Find,  by  adding,  the  number 
of  hours  in  one  week.     Multiply  and  prove  your  answer  correct. 

5.  One  square  mile  contains  640  A.  Find,  by  adding,  the  num- 
ber of  acres  in  6  sq.  mi.     Multiply  and  prove  your  answer  correct. 

6.  One  gallon  contains  231  cu.  in.  Show,  by  adding,  that  a 
5-gallon  can  contains  1155  cu.  in.     Show  this  by  multiplication. 

7.  Find  the  number  of  days  in  6  yr.  of  365  da.  each  and  2  leap 
years  of  366  da.  each. 

8.  If  a  boy  attends  school  189  da.  each  year,  find,  by  adding,  the 
number  of  days  he  will  be  in  school  in  8  yr.  Multiply  and  prove 
your  answer  correct. 

Exercise  30 

In  each  of  the  following  questions  state  in  each  case  which  is 
the  whole  quantity  to  be  measured  and  what  are  the  parts  meas- 
uring it. 

1.  A  spent  the  following  sums  of  money:  $425,  $342,  $673, 
and  $  897.     How  much  did  he  spend  all  together  ? 

2.  An  encyclopaedia  consists  of  three  volumes.  In  the  first 
there  are  693  pages,  in  the  second  745,  and  in  the  third  892.  Find 
the  number  of  pages  in  the  encyclopaedia. 

3.  Using  the  table  in  §  150,  find  the  number  of  days  in  the 
first  six  months  of  the  year.     In  the  last  six  months.     In  a  year. 


42  ARITHMETIC 

4.  Find  the  number  of  days  in  the  three  spring  months.  In 
the  three  summer  months.  In  the  three  fall  months.  In  the 
three  winter  months. 

5.  Find  the  total  area  of  these  lakes  : 

Lake  Erie,  area    7,750  sq.  mi. 

Lake  Ontario,  area  6,950  sq.  mi. 
Lake  Michigan,  area  22,000  sq.  mi. 
Lake  Superior,   area  31,500  sq.  mi. 

6.  A  merchant  bought  150  yd.  of  cloth  for  $232,  254  yd.  for 
$175,  1875  yd.  for  $2395,  and  640  yd.  for  $1966.  Find  the 
number  of  yards  bought  and  the  total  cost. 

7.  How  many  years  are  there  between  the  establishment  of 
the  Kepublic  of  Rome  in  509  b.c.  and  the  Declaration  of  Inde- 
pendence in  1776  A.D.  ? 

8.  Find  the  sum  of  three  hundred  seventy-six  thousand  fifty- 
four  ;  one  hundred  ninety-seven  thousand  two  hundred  fifty-one ; 
four  hundred  fifty-seven  thousand  six  hundred  forty-nine. 

9.  The  population  of  Maine  is  694,466;  New  Hampshire, 
411,588;  Vermont,  343,641;  Massachusetts,  2,805,346;  Rhode 
Island,  428,556 ;  and  Connecticut,  908,355.  Find  the  population 
of  the  New  England  states. 

10.  According  to  the  census  of  1900  the  population  of  the  six 
largest  cities  of  the  United  States  is :  New  York,  3,437,202 ; 
Chicago,  1,698,575 ;  Philadelphia,  1,293,697 ;  St.  Louis,  575,238 ; 
Boston,  560,892 ;  and  Baltimore,  508,957.  Find  the  total  popu- 
lation. 

11.  Find  the  number  of  times  a  clock  strikes  from  :i  (piartor 
of  nine  a.m.  until  a  quarter  of  nine  p.m. 

12.  A,  B,  and  C  engaged  in  trade;  A  put  in  $3475,  B  $4593, 
and  C  as  much  as  the  other  two  together.  How  much  money  was 
put  into  the  business  ? 


additio:n^ 


43 


13.  A  man,  dying,  willed  to  his  widow,  $6875;  to  his  son, 
$  4294,  and  to  his  daughter,  $  3875.     What  was  his  estate  worth  ? 

14.  According  to  the  census  of  1900  the  population  of  the 
following  states  is:  Wisconsin,  2,069,042;  Illinois,  4,821,550; 
Indiana,  2,516,462  ;  Ohio,  4,157,545 ;  Michigan,  2,420,982.  Find 
their  total  population. 

15.  The  area  of  the  basin  of  the  Colorado  Eiver  is  250,000 
sq.  mi.;  Columbia,  250,000 ;  Mackenzie  River,  440,000;  Missouri- 
Mississippi,  1,250,000;  Nelson,  355,000;  Rio  Grande,  180,000; 
St.  Lawrence,  350,000.     Find  the  total  area  of  these  river  basins. 

16.  What  is  the  area  of  the  New  England  states;  that  of 
Maine  being  in  square  miles  33,040,  of  New  Hampshire  9305,  of 
Vermont  9565,  of  Massachusetts  8315,  of  Rhode  Island  1256,- 
of  Connecticut  4990  ? 

17.  A  father  left  his  eldest  son  $  24,000  more  than  he  left  his 
second  son,  and  the  second  son  $  7560  more  than  the  third ;  to 
the  third  he  left  $  60,480.  What  was  the  second  son's  portion  ? 
What  was  the  portion  of  the  oldest  son  ? 

18.  Aug.  17,  1899,  there  were  inspected  in  the  city  of  Chicago 
66  carloads  of  wheat,  281  of  corn,  413  of  oats,  3  of  rye,  and  30 
of  barley.     Find  the  total  number  of  carloads  of  grain  inspected. 

19.  For  the  week  ending  Aug.  17,  1899,  there  was  received  in 
Chicago  the  following  number  of  live  stock.  Find  the  total 
number  of  each  kind: 


Cattle 

Calves 

Hogs 

Sheep 

Thursday,  Aug.  10 

10,122 

374 

23,007 

14,565 

Friday,  Aug.  11  .     . 

3,342 

147 

16,113 

9,017 

Saturday,  Aug.  12    . 

100 

10 

8,614 

2,171 

Monday,  Aug.  14     . 

19,005 

293 

20,466 

28,960 

Tuesday,  Aug.  15     . 

5,960 

1,488 

12,496 

18,860 

Wednesday,  Aug.  16 

23,869 

565 

23,697 

28,134 

Thursday,  Aug.  17  .     .     .     . 

10,500 

300 

25,000 

15,000 

44 


ARITHMETIC 


20.    The  imports  to  the  United  States  from  Cuba,  Porto  Rico, 
and  the  Philippines,  from  Jan.  1  to  July  31,  1899,  were : 


Cuba 


Porto  Rico 


Philippines 


January 
February 
March  . 
April  . 
May 
June  . 
July      . 


$994,220 
2,307,940 
3,398,723 
4,419,712 
4,762,970 
3,614,904 
2,632,845 


$63,481 
124,618 
349,785 
782,172 
647,179 
814,803 
448,267 


$348,019 
277,003 
147,452 
937,164 
622,101 
61,882 
880,515 


Find  the  total  in  each  case. 

21.    The  exports  from  the  United  States  to  Cuba,  Porto  Rico, 
and  the  Philippines,  from  Jan.  1  to  July  31,  1899,  were : 


January 
February 
March  . 
April     . 
May 
June 
July      . 


Cuba 


$1,980,982 
1,671,846 
2,503,110 
1,723,062 
2,124,679 
2,123,935 
1,989,379 


POETO    Rico 


$224,150 
267,619 
375,529 
316,669 
305,564 
361,423 
448,267 


Philippini 


$  15,382 
19,529 
43,180 

112,267 
63,905 
67,775 
64,408 


Find  the  total  in  each  case. 


22.  Dec.  1,  1899,  there  were  905  officers  and  30,578  men  of  the 
regular  army  of  the  United  States  in  the  Philippines,  and  594 
officers  and  15,388  men  of  the  volunteer  force.  Find  the  total 
number  of  officers  and  of  men  in  the  Philippines  Dec.  1,  1899. 

23.  Dec.  1,  1899,  the  regular  army  of  the  United  States  was 
distributed  as  follows: 


ADDITION 


45 


Officers 

Enlisted  Men 

In  Cuba     .               

334 

87 
910 

12 
905 

10,796 

Tn  Porto  Rico 

2,855 

17,317 

453 

On  the  continent  of  North  America  .     . 
In  Hawaii 

In  the  Philippine  Islands 

30,578 

Find  the  number  of  officers  and  of  enlisted  men  in  the  army. 

24.    In   five   months   of  the   year   1899,  a  rural   mail-carrier 
handled  the  following  mail  matter: 


Pieces  Collected 


June     .     . 
July      .     . 
August 
September 
October     . 


2086 

2233 

746 

890 

930 


Find  the  total  number  of  pieces  delivered  and  collected  each  month. 

25.    Add  vertically  and  horizontally  the  following  statement 
of  eight  weeks'  cash  receipts: 


MON. 

TUES. 

Wed. 

Thur. 

Fri. 

Sat. 

Total 

1st 

$3862.93 

$1391.76 

$6760.68 

$1098.91 

$1696.65 

$43.68 

2d 

396.74 

6168.37 

864.39 

964.26 

167.69 

1864.86 

3d 

1768.63 

467.89 

2035.68 

3165.03 

691.83 

785.97 

4th 

3976.98 

76.05 

364.76 

93.68 

1948.39 

1759.46 

5th 

263.76 

1035.84 

36.10 

386.41 

3.45 

1396.71 

6th 

1559.83 

1932.57 

1268.15 

8.37 

279.72 

67.85 

7th 

62.24 

318.62 

134.36 

1763.29 

1468.29 

543.66 

8th 

194.87 

3.85 

7643.82 

685.38 

765.42 

39.67 

Total 

46  ARITHMETIC 

26.   Add  vertically  and  horizontally  the  following  statement : 


Total 


$1169.84 

909.58 

575.72 

2678.28 

312.83 

1052.47 

339.11 

1732.50 

1237.50 

113.56 

3661.00 

1139.07 


$3650.12 

866.78 

742.49 

1180.66 

16.38.24 

342.65 

687.23 

514.02 

3839.25 

1291.98 

973.03 

670.22 


$189.10 
914.19 

1654.70 
119.25 

2016.72 
108.00 
215.17 
557.60 
777.60 
112.50 
311.20 

1201.64 


$97.22 
239.49 
196.17 

8418.60 

1542.24 
349.95 

1020.00 
600.00 
136.70 

1850.14 
636.99 

7357.51 


$26.55 
297.02 
869.69 

2223.42 

5300.20 
136.97 

1124.50 
475.00 

4656.65 
738.75 
243.44 
252.47 


$851.02 

312.60 

1477.42 

668.35 

116.02 

1214.03 

1732.25 

138.60 

1097.47 

1204.74 

142.91 

694.62 


Total 


25.   Find  the  sum  of:  2.46,  23.973,  15.025,  643.319,  and 

,468. 


2.46 

23.973 

15.025 

643.319 

.468 

685.245 


Since  we  can  add  numbers  of  the  same  unit,  we  write  the 
addends  so  that  units  will  be  under  units,  tenths  under  tenths, 
and  so  on.  This  is  easily  done  by  placing  the  decimal  points 
directly  below  each  other.  Then,  beginning  at  the  right,  we 
add  the  figures  as  if  they  were  integers,  and  place  the  decimal 
point  in  the  sura  between  the  units'  and  tenths'  column. 


Add: 
1. 


. 

Exercise  31 

3.456 

2.  27.43 

4.593 

18.314 

7.245 

5.687 

9.864 

34.986 

3.124 

22.426 

3. 


76.425 
39.639 
28.764 
21.385 
13.026 


ADDITION  47 

Write  in  columns  and  add  : 

4.  4.396  +  7.295  +  6.478  +  5.765. 

5.  .432  +  .987  +  .593  +  .666. 

6.  84.63  +  46.892  +  24.7  +  95.657. 

7.  $24,375  +  $95,875  +  $16,125  +  $19.50. 

8.  Find  the  capacity  of  four  bins,  the  first  of  which  contains 
66.384  bu.,  the  second  89.645  bu.,  the  third  27.437  bu.,  and  the 
fourth  75.938  bu. 

9.  What  is  the  area  of  a  farm  which  is  divided  into  three 
fields  containing,  respectively,  25.936  A.,  14.56  A.,  and  24.504  A.? 

10.  Four  towns,  A,  B,  C,  D,  lie  on  a  road  running  directly  east 
and  west.  The  distance  from  A  to  B  is  5.693  mi.,  from  B  to  C 
8.421  mi.,  from  C  to  D  12.768  mi.     Find  the  distance  from  A  to  D. 

11.  In  the  city  of  Chicago  in  1899  there  were  243.57  mi.  of 
sidewalk  built  and  67.88  mi.  repaired.  Find  the  total  number 
built  and  repaired. 

Miscellaneous  Exercise  32 

1.  A  boy  worked  20  problems  in  arithmetic  and  got  wrong 
answers  to  ^^  of  them.  How  many  did  he  miss  ?  How  many 
did  he  work  correctly  ? 

2.  Caryl  had  a  spelling  lesson  of  48  words,  and  spelled  cor- 
rectly I  of  them.     How  many  did  she  spell  correctly  ? 

3.  During  the  month  of  June  a  commercial  traveller  was 
away  from  home  f  of  the  time.     How  many  days  was  he  at  home  ? 

4.  What  is  the  ratio  of  8  lb.  to  16  lb.  ?  Of  the  cost  of  12  lb. 
of  sugar  to  that  of  15  lb.  ?  Of  15  lb.  to  12  lb.  ?  Of  the  weight  of 
36  yd.  of  carpet  to  27  yd.  of  the  same  kind  ? 

5.  A  pole  6  ft.  high  casts  a  shadow  8  ft.  long.  By  what  must 
8  ft.  be  multiplied  to  find  the  length  of  the  shadow  cast  by  a  pole 
9ft.  long?     How  long  is  it? 

6.  The  carpet  on  a  certain  room  weighs  35  oz.  to  the  yard, 
and  its  entire  weight  is  105  lb.  What  would  have  been  its  weight 
if  it  had  been  made  out  of  carpet  weighing  45  oz.  to  the  yard  ? 


48 


ARITHMETIC 


7.  What  is  the  ratio  of  75  lb.  to  90  lb.?  A  carpet  weighs 
30  oz.  to  the  yard,  and  its  entire  weight  is  90  lb.  Another  carpet 
of  the  same  size  weighs  75  lb.  Find  how  many  ounces  1  yd.  of 
the  second  carpet  weighs. 

8.  Find  the  cost  of  2  yd.  of  cloth  at  32  ^  a  yard.  Find  the 
cost  of  3  doz.  oranges  at  $  .25  a  dozen. 

9.  Find  the  selling  price  of  4  bu.  of  wheat  at  64^  a  bushel. 
Find  the  selling  price  of  5  bu.  of  corn  at  $  .38  a  busheh 

10.  Find  the  cost  of  12  yd.  of  oil  cloth  at  $  .21  a  yard.  Of 
18  yd.  at  $  .25  a  yard. 

11.  What  is  20%  of  $30?  12^%  of  $24?  37^%  of  f  72? 
33i%of$69?  6i%of$32?  62i%of$48?  lli%  of  $54? 
16|%of$54?  66|%of$21?  8^%  of  $36?  75%  of  $84? 
6  J  %  of  $  45  ?     What  is  meant  by  saying  that  6|%  =yV  ? 

12.  A  grocer  bought  tea  at  75  ^  a  pound,  and  sold  it  at  a  gain 
of  33^%.     Find  his  gain.     Find  the  selling  price  per  pound. 

13.  How  do  you  find  the  gain  when  you  are  given  the  cost  and 
the  gain  per  cent  ?     How  do  you  find  the  selling  price  ? 

14.  In  the  following  examples,  find  the  gain  and  also  the  selling 
price : 


Cost 

Gain  Per  Cent 

iOf 

i2r/o 

Z^f 

^^% 

$  .65 

20% 

$2.60 

25% 

$2.10 

6r/o 

%   .28 

14if% 

%   .48 

i«r/o 

%   .16 

371% 

%   .55 

9iV% 

$2.19 

66|% 

$1.00 

75% 

$1.32 

85% 

$   .63 

22i% 

$    .08 

100  % 

Gain 


Selling  Prick 


ADDITION  49 

15.  A  grocer  bought  berries  at  8  )^  a  box  and  sold  them  at  a 
gain  of  25%.  Find  the  selling  price.  What  did  he  receive  for  a 
crate  containing  24  boxes  ? 

16.  A  grocer  bought  berries  at  6  ^  a  box  and  sold  them  at  a 
gain  of  33^%.     What  did  he  receive  for  a  crate  of  16  boxes  ? 

17.  In  the  year  1899  the  United  States  sold  to  the  British 
Colonies  the  following  articles : 

Cotton $2,994,674 

Corn        . 7,501,508 

Wheat 6,159,136 

Flour 9,961,230 

Provisions 16,886,946 

Refined  petroleum 4,211,709 

Live  cattle 701,947 

Tobacco 1,251,407 

Find  the  total  value. 

18.  Add: 

32.49  144.938  92.778 

63.824  992.807  89.006 

765.653  25.069  729.054 

4.159  76.238  308.298 

36.768  4.718  123.864 


CHAPTER  V 

SUBTRACTION 

26.  A  man  who  earned  $14  a  week,  spends  15  a  week  for 
his  board.     How  much  has  he  left  ? 

We  are  here  given  the  whole  quantity,  or  il4,  and  one 
part,  or  $5,  and  we  are  required  to  find  the  otlier  part. 

The  question  may  be  viewed  in  two  ways  :  How  much 
must  be  added  to  $5  to  make  §14  ?  Or  how  much  must  be 
taken  from  1 14  to  leave  i5?  The  answer  in  both  cases  is 
known  from  addition.  $5  and  §9  are  two  quantities  making 
$14.  Therefore,  if  one  of  them,  i5,  is  given,  the  other  must 
be  $9.  Or,  in  other  words,  $9  is  the  difference  between  $14 
and  $5.  It  is  this  view  of  difference  that  gives  the  name 
Subtraction, 

27.  Subtraction  may  therefore  be  defined  as  the  operation 
of  finding  the  part  of  a  given  quantity  that  remains  when  a 
given  part  has  been  taken  from  the  quantity. 

The  given  quantity  is  called  the  Minuend,  and  the  given 
part  the  Subtrahend,  while  the  part  that  remains  is  called  the 
Difference  or  Remainder. 

28.  The  Sign  of  Subtraction,  — ,  is  called  minus.  Thus 
8  —  6  is  read  8  minus  6,  and  signifies  that  6  is  to  be  sub- 
tracted from  8. 

Exercise  33 

Read  the  following  questions,  filling  in  the  blanks : 

1.  6  and  7  are  — ,       8  and  9  are  — ,     4  and  8  are  — . 

2.  4  and  6  are  — ,       4  and  —  are  10,  9  and  —  are  15. 

60 


SUBTRACTION  61 

3.  2  and  —  are  11,     3  and  —  are  8,     6  and  —  are  14. 

4.  22  and  —  are  25,   4  and  —  are  36,  9  and  —  are  27. 

5.  8  and  —  are  29,      5  and  —  are  16,  6  and  —  are  48. 

Subtract  (Note :  Let  the  process  be  not  6  from  9  leaves  3,  but 
6  and  3  are  9) : 

9     19     29    39    49     69     99 
6'     6'     6'     6'     6'     6'     6 
90.  190.  8.  28.  80.  380 
60'     60'  5'     5'  50'     50* 
12.  120.  32.  320.  14.  54 
•     7'     70'     7'     70'     8'     8 

What  numbers  added  respectively  to  9,  7,  6,  8,  5,  and  4,  make 
9.    12?      10.    15?      11.    17?      12.    14?      13.    18?      14.    16? 

29.  Drill,  as  in  §  21  in  addition,  on  the  fundamental 
subtractions,  connecting  with  corresponding  additions,  until 
accuracy  and  rapidity  are  secured  ;  thus  : 


,     ^,  —-,  — -,  —-,  -— ;  and  so  on. 


9.  19.  29.  37    49     59 

8'     8'     8' 

90     190     210    390     490     „    .       _ 

— - ;  ;   :  ;  ;  and  so  on. 

80'     80'     80'     80      80' 

17    27     36    47     57         . 
■8'^'^'^'-8'""^'''" 

30.   A  man  who  owned  18  farms  of  50  A.,  sold  7  of  them. 
How  much  had  he  left  ? 

Because  7  +  11  =  18,  it  is  evident  that  he  had  left  11  farms  of  50  A. 
each. 

Exercise  34 

1.  9  ft. +?  =  16ft.     9  yd. +?  =  16yd.     12  yd.  -  4  yd.  =  ? 

2.  How  many  dimes  must   be  added   to  6  dimes  to  get  14 
dimes  ?     What  is  the  difference  between  14  dimes  and  6  dimes  ? 


52  arithmp:tic 

How  much  less  is  6  five-dollar  bills  than  14  five-dollar  bills? 
How  many  dollars  ? 

3.  A  fruit  dealer  arranges  his  oranges  into  12  piles  of  4 
oranges  each.  He  sells  8  piles.  How  many  has  he  left  ?  How 
many  oranges  ? 

4.  A  fruit  dealer  sold  10  piles  of  3  apples  each.  How  many 
had  he  left  if  he  had  at  first  15  piles  of  3  apples  ?  How  many 
apples  ? 

5.  I  owe  a  debt  of  12  ten-dollar  bills  and  have  5  ten-dollar 
bills  in  my  pocket.  If  I  pay  this  toward  the  debt,  how  much 
do  I  still  owe  ?     How  many  dollars  ? 

6.  What  must  be  added  to  15  units  to  get  20  units  ?  Taken 
from  20  units  to  get  15  units  ?  To  get  5  units  ?  If  I  sell  my 
horse  for  20  units  of  $  5  each,  find  the  selling  price. 

7.  What  is  the  difference  between  a  quantity  denoted  by  14 
times  the  measuring  unit  and  one  denoted  by  8  times  the  meas- 
uring unit  ? 

8.  A  person  who  has  $50  pays  a  debt  of  $30.  How  much 
money  has  he  left  ?  If  this  is  in  ten-dollar  bills,  how  many  ?  If 
in  five-dollar  bills,  how  many  ? 

9.  3  gal.  =  ?  qt.  5  gal.  =  ?  qt.  8  gal.  =  ?  qt.  From  a  can 
containing  2  gal.  of  milk  a  milkman  poured  3  qt.  How  many 
quarts  were  left  in  the  can? 

10.  8  qt.  =  ?  pt.  12  qt.  =  ?  pt.  27  qt.  =  ?  pt.  A  lady  bought 
16  qt.  of  fruit  in  pint  jars.  How  many  jars  were  left  after  13 
had  been  used  ? 

11.  How  many  quarts  in  1  bu.  ?  In  2  bu.  ?  A  bushel  basket 
lacked  5  qt.  of  being  full.     How  many  quarts  did  it  contain  ? 

12.  A  piece  3  ft.  4  in.  long  is  cut  from  a  rope  8  ft.  6  in.  long. 
Find  the  length  of  the  other  piece. 

13.  In  paying  for  a  suit  of  clothes  that  cost  f  16  I  gave  the 
clerk  2  ten-dollar  bills.     What  change  should  I  receive  back  ? 


SUBTRACTION  53 

14.  1  sq.  ft.  =  ?  sq.  in.  A  piece  of  cloth  8  in.  long  and  6  in. 
wide  is  cut  from  a  square  foot  of  cloth.  How  many  square  inches 
in  the  remainder  ? 

15.  2  ft.  =  ?  in.  4  ft.  =  ?  in.  5  ft.  =  ?  in.  A  man  takes  a 
step  3  ft.  long  and  his  son  a  step  8  in.  shorter.  How  many 
inches  does  his  son  step  ? 

16.  Out  of  6  doz.  eggs  half  a  dozen  were  found  to  be  broken. 
How  many  were  whole  ? 

31.  The  following  method  of  subtraction,  which  is  nearly 
always  adopted  in  making  change,  is  almost  universally 
employed  by  professional  computers  and  is  considered  by 
many  teachers  the  best  way  to  perform  subtraction. 

It  is  superior  in  accuracy  and  rapidity  to  the  method  of 
the  next  paragraph. 

It  is  based  on  the  principle  that  the  sum  of  the  subtrahend 
and  remainder  is  equal  to  the  minuend. 


From  875  take  451.  Thus :  l  and  4  are  5  ;  5  and  2  are  7  ;  4  and 

orrr  4  are  8.     In  this  operation  let  the  pupil  fancy 

that  he  is  doing  addition  with  the  sum  at  the 
top,  and  as  he  works  set  down  the  figures,  4,  2, 


451 


424  and  4. 

32.  A  merchant  bought  965  yd.  of  silk  and  sold  723  yd. 
How  much  had  he  left  ? 

Here  we  are  required  to  find  the  unknown  part.  This  is  the  difference 
between  the  measured  whole,  or  965  yd.,  and  the  given  part,  723  yd. 

965  yd.  =  the  measured  whole. 

723  yd.  =  the  measured  part. 

242  yd.  =  the  difference,  which  is  now  definitely  known. 

Explanation.  — As  in  addition,  we  write  units  under  units^  tens  under 
tens,  and  hundreds  under  hundreds.  Beginning  with  the  units,  we  say  3  units 
from  5  units  leaves  2  units,  which  we  write  below  the  line  in  the  units'  column. 
Then  2  tens  from  6  tens  leaves  4  tens.     Place  this  in  the  tens'  column. 


54  ARITHMETIC 

Lastly,  7  hundreds  from  9  hundreds  leaves  2  hundreds,  which  we  write  in  the 
hundreds'  column. 

This  difference,  242  yd. ,  is  the  other  part,  which  is  now  definitely  measured. 

Exercise  35 

Subtract,  and  prove  your  answer  correct  in  each  case : 

1.  946        2.  785         3.  659        4.  897 
324  323  236  683 

5.  8498        6.  9999        7.  8395        8.  7948 
2361  7265  4073  5216 

9.  7684       10.  8697       11.  2578       12.  3796 
6450  1082  1506  1542 

Exercise  36 

In  the  following  questions,  name  (1)  the  unknown  part,  (2)  the 
whole  quantity,  (3)  the  given  part : 

1.  A  merchant  sold  246  yd.  from  a  piece  of  cloth  258  yd.  in 
length.     How  many  yards  had  he  remaining  ? 

2.  A  person  deposited  in  a  bank  ^8495,  but  shortly  after  drew 
out  $1035.  How  much  had  he  left  in  the  bank?  If  he  should 
draw  out  this  sum  in  ten-dollar  bills,  how  many  would  he  get  ? 

3.  On  Tuesday  a  merchant  deposited  in  a  bank  $3475,  on 
Wednesday  $4690.  If  he  withdi;ew  $1010  on  Thursday,  how 
much  did  he  still  have  on  deposit  ? 

4.  What  is  the  difference  between  1  yr.  and  213  da.  ? 

5.  A  bankrupt  has  debts  amounting  to  $8496;  his  assets  are 
$3015.     How  much  more  does  he  owe  than  he  can  pay  ? 

6.  A  man  left  property  to  the  value  of  $36,875  to  his  two 
children.  The  son  received  $  14,250 ;  what  was  the  daughter's 
share  ? 

7.  At  an  election  the  successful  candidate  received  953  votes, 
and  the  unsuccessful  candidate  613  votes.  Find  the  majority  of 
the  former. 


SUBTRACTION  55 


33.    Computers'  Method. 


From  94,275  take  67,492.  Thus :  2  and  3  are  5  ;  9  and  8  are  17  ; 

Q497  ^  carry  1  to  4  as  in  addition,  making  it  5 ; 

5  and  7  are  12  ;  carry  1  to  7,  making  it 

^^^^^  8 ;  8  and  6  are  14  ;  carry  1  to  6,  making 

26783  it  7  ;  7  and  2  are  9. 

The  numbers  3,  8,  7,  6,  and  2  are  written  down  in  order  to  give  the 

remainder. 

34.   Find  the  difference  between  642  and  375. 

As  we  cannot  take  5  units  from  2  units,  we  take  1  ten  from  the  4 
^  tens,  and  adding  this  1  ten,  which  equals  ten  units,  to  the  2  units,  we 

375  have  12  units.  Then  5  units  from  12  units  leaves  7  units,  which  we 
267  write  under  the  units'  column.  Now  as  we  took  1  ten  from  4  tens, 
we  have  left  only  3  tens  ;  we  borrow  1  hundred  from  the  6  hundreds, 
and  considering  the  1  hundred  as  10  tejis,  we  add  it  to  the  3  tens,  making  13 
tens  ;  then  7  tens  from  13  tens  leaves  6  tens,  which  we  write  under  the  tens' 
column. 

Now  as  we  took  1  hundred  from  6  hundreds,  we  have  left  only  5  hundreds ; 
hence  we  subtract  3  hundreds  from  5  hundreds,  leaving  only  2  hundreds, 
which  we  write  in  the  hundreds'  column. 

The  remainder,  or  difference,  is  thus  2  hundreds,  6  tens,  and  7  units,  or  267. 

Exercise  37 

■"    *  In  the  following  questions  subtract  and  prove  the  correctness 
of  your  results  by  adding  the  two  parts : 

1.  653  2.  307 

269  268 

4.  921  5.  255 

87  99 

7.  3849  8.  9345 

2567  -   8367 


10.  8000  11.  9041 

5348  7385 


3. 

642 

375 

6. 

907 

859 

9. 

7007 

6609 

2. 

7968 

2693 

Use  computers'  method  of  subtraction. 


56 

ARITHMETIC 

13.  43970 

14.  50062 

15 

.  12009 

26784 

37891 

11376 

16.  34060 

17.  986403 

18 

.  620703 

29143 

728547 

444444 

19.  850439 

20.  759826 

473642 

378934 

*  Subtract : 

Exercise  38 

1.  57261 

2.  40359 

3. 

10000 

38877 

9998 

1021 

4.  89437 

5.  671*2 

6. 

81349 

15790 

30293 

47538 

7.  654375 

8.  986392 

9. 

303233 

412884 

826957 

192001 

10.  233826 

11.  310865 

12. 

605487 

204739 

270326 

584598 

13.  164326 

14.  982623 

15. 

1000101 

48476 

897674 

707707 

*  Subtract : 

Exercise  39 

1.  755903 

2.  640021 

3. 

716287 

699004 

400569 

662763 

4.  100794 

5.  143812 

6. 

948735 

81685 

109758 

473596 

7.  4731246 

8.  9487352 

9. 

1737682 

4342760 

5999999 

739908 

Use  computers'  method  of  subtraction. 


SUBTRACTION 


57 


10.  3801572 
2003789 

13.  1217191 
1038182 

16.  5468305 
1490673 

19.  8235460 
3530089 


11.  5745861 
2837154 

14.  4100293 
1925867 

17.  7086543 
2889454 

20.  2679953 
1346397 


12.  5048650 
4243091 

15.  2047000 
1054888 

18.  1671498 
536819 


21.  1521815 
1432568 


Exercise  40 

1.  1  bu.  =  ?  qt. 

Subtract  32  qt.  from  128  qt.,  and  from  the  remainder,  and  so 
on,  until  no  remainder  is  left.  How  many  times  did  you  sub- 
tract ?     How  many  bushels  in  128  qt.  ? 

2.  Divide  128  qt.  by  32  qt.,  and  thus  show  that  your  answer  to 
example  1  is  correct. 

3.  1  lb.  =  ?  oz. 

As  in  example  1,  subtract  16  oz.  from  80  oz.  until  no  remainder 
is  left.  How  many  pounds  in  80  oz.  ?  Prove  your  answer  correct 
by  division. 

4.  1  da.  =  ?  hr. 

As  in  example  1,  subtract  24  hr.  from  168  hr.  until  no  re- 
mainder is  left.  How  many  days  in  168  hr.  ?  Prove  your  answer 
correct  by  division. 

5.  A  township  contains  36  sq.  mi. 

As  in  example  1,  subtract  36  sq.  mi.  from  216  sq.  mi.  until  no 
remainder  is  left.  Into  how  many  townships  can  a  section  of 
country  containing  216  sq.  mi.  be  divided  ?  Prove  your  answer 
correct  by  division. 

6.  1  sq.  ft.  =  144  sq.  in. 

As  in  example  1,  subtract  144  sq.  in.  from  1152  sq.  in.  until 
no  remainder  is  left.  How  many  square  feet  in  1152  sq.  in.? 
Prove  your  answer  correct  by  division. 


68  ARITHMETIC 

7.  1  mi.  =  1760  yd. 

As  in  example  1,  subtract  1760  yd.  from  8800  yd.  until  no 
remainder  is  left.  How  many  miles  in  8800  yd.  ?  Prove  by 
division. 

8.  1  mi.  =  5280  ft. 

As  in  example  1,  subtract  5280  ft.  from  31,680  ft.  until  no 
remainder  is  left.  How  many  miles  in  31,680  ft.  ?  Prove  by 
division. 

Exercise  41 

Solve  the  following  questions  and  prove  your  answers  correct : 

1.  Subtract  f  819  from  .$918,  explaining  the  process. 

2.  I  paid  $3500  for  a  house  and  lot  and  sold  it  for  $4275. 
Find  my  gain. 

3.  A  speculator  sold  cattle  at  a  loss  of  $  3145  and  some  horses 
at  a  gain  of  $  2578.    How  much  did  he  lose  on  both  transactions  ? 

4.  A  merchant  exchanges  a  stock  of  goods  worth  f  6725,  and 
a  house  worth  $  3120,  with  a  farmer  for  a  farm  valued  at  $  5900, 
the  farmer  paying  the  balance  in  money.  What  sum  must  the 
merchant  receive  ? 

5.  A  lends  B  $9780;  B  repays  A  by  giving  him  bank  stock 
to  the  amount  of  $  1946,  a  farm  worth  $  6385,  and  the  balance  in 
cash.     How  much  cash  did  B  pay  A  ? 

6.  A  is  worth  $  6215,  B  is  worth  $  876  less  than  A,  and  C  is 
worth  as  much  as  A  and  B  together,  lacking  $  2343.  How  much 
are  B  and  C  worth,  respectively  ?    How  much  are  all  three  worth  ? 

7.  How  much  larger  is  Lake  Erie  than  Lake  Ontario?  Lake 
Superior  than  Lake  Michigan?  The  total  areas  of  the  three 
smaller  lakes  than  Lake  Superior  ?  (For  the  areas  of  these  lakes 
see  Exercise  30,  question  5.) 

8.  During  August,  1899,  the  postal  receipts  at  Chicago 
amounted  to  $495,093,  an  increase  of  $37,714  over  the  receipts 
for  the  same  month  of  1898.  The  receipts  at  Detroit  in  August 
last  were  $  53,238,  a  decrease  of  $  239  as  compared  with  August, 


SUBTRACTION  59 

1898 ;  at  Minneapolis,  $  53,201,  an  increase  of  $  1756.     Find  the 
postal  receipts  in  ea6h  of  these  cities  for  August,  1898. 

9.  What  is  the  ditference  between  9  and  5?  What  quantity 
is  equal  to  this  difference  if  the  unit  of  measure  is  3  in.  ?  8  in.  ? 
$  5  ?     $  10  ? 

10.  What  is  the  difference  between  643  and  579  when  the  unit 
of  value  is  $  1  ?     $  10  ?     ^  100  ?     $  1000  ? 

11.  How  much  greater  are  253  units  of  f  1000  than  1864  units 
of  f 100  ? 

12.  A  man  bought  a  house  and  lot  for  $8450.  He  spent 
$  1379  in  improvements  and  $  212  for  insurance.  He  then  sold 
the  house  and  lot  for  $  12,000 ;  did  he  gain  or  lose,  and  how 
much? 

13.  A  collector  received  $1300  from  five  men;  from  the  first 
he  received  $  367,  from  the  second  $  194  less  than  from  the  first, 
from  the  third  $36  more  than  from  the  second,  from  the  fourth 
as  much  as  from  the  second  and  third  together.  How  much  did 
he  collect  from  each  man  ? 

14.  From  the  difference  between  784  and  8305,  take  the  differ- 
ence between  17,012  and  21,410. 

15.  Two  men  start  from  the  same  point  and  travel  in  the 
same  direction.  The  first  travels  84  mi.  in  one  day  and  the 
second  69  mi.  How  far  were  they  apart  at  the  end  of  the  first 
day  ?  If  they  had  travelled  in  opposite  directions,  how  far  would 
they  have  been  apart  ? 

16.  The  population  of  Texas  in  1900  was  3,048,710,  and  of 
Illinois,  4,821,550.  How  much  greater  was  the  population  of 
Illinois  in  1900  than  that  of  Texas  ? 

17.  The  area  of  Texas  is  265,780  sq.  mi.,  of  England,  50,800 
sq.  mi.,  and  of  Germany,  208,700  sq.  mi.  How  much  larger  is 
Texas  than  the  united  area  of  England  and  Germany  ? 

18.  The  population  of  New  York  State  in  1880  was  5,082,871 ; 
in  1890  it  was  5,997,853 ;  in  1900  it  was  7,268,012.     What  was 


60 


ARITHMETIC 


the  increase  in  population  from  1880  to  1890?  From  1890  to 
1900?  How  much  greater  was  the  latter  increase  than  the 
former  ? 

19.  The  population  of  Illinois  in  1900  was  4,821,550;  of  Iowa, 
2,231,853.  Show  by  subtraction  that  the  population  of  Illinois  in 
1900  was  more  than  twice  that  of  Iowa. 

20.  A  Frenchman  came  to  the  United  States  June  20,  1844, 
when  he  was  23  years  old.     How  old  was  he  June  20,  1899  ? 

21.  Find  the  increase  or  decrease  in  the  earnings  of  the  follow- 
ing railroads  for  the  first  week  of  August,  1899,  over  the  first  week 
of  August,  1898 : 


Pittsburg  &  Western : 

First  week  August 
Norfolk  &  Western : 

First  week  August 
Ohio  River : 

First  week  August 
Rio  Grande  Western : 

First  week  August 


1899 


$64,934 

270,086 

28,689 

60,800 


$65,115 

215,392 

22,780 

44,600 


22.  The  area  of  winter  wheat  sown  in  Iowa  in  the  fall  of  1898 
was  154,177  A.  On  account  of  winter-killing  only  27,427  A. 
were  harvested.  Find  the  number  of  acres  of  wheat  destroyed 
by  the  cold. 

23.  154,243  A.  were  planted  in  potatoes  in  Iowa  in  1899,  and 
155,131  A.  in  1898.     Find  the  decrease  in  1899. 

24.  The  United  States  government  paid  $30,393,209.53  for 
carrying  the  mails  in  1888,  and  $552,294,383.23  in  1898.  Find 
the  increase. 

25.  The  imports  into  the  United  States  from  Cuba,  Porto 
Rico,  and  the  Philippines  for  the  seven  months  ending  July  31, 
1898  and  1899,  are  given  below.  Find  the  increase  in  each  case 
for  the  year  1899. 


SUBTRACTION 


61 


Imports 

Cuba 

Porto  Rico 

Philippines 

1899 

1898   ..... 

$  19,976,956 
12,474,770 

$3,379,944 
2,253,800 

$3,274,134 

2,283,775 

26.    As  in  the  previous  example,  find  the  increase  in  exports 
from  the  United  States  to  these  islands. 


Exports 

Cuba 

Porto  Rico 

Philippines 

1899 

1898 

$14,116,993 
4,485,937 

$2,299,221 
569,110 

$386,109 
65,736 

35.  From  25.3846  take  18.6397. 

We  write  units  under  units,  tenths  under  tenths,  and  so 


25.3846 
18.6397 

6.7449 


on.  Beginning  at  the  right,  we  subtract  as  if  the  figures  were 
integers,  and  place  the  decimal  point  in  the  difference  between 
the  units'  and  the  tenths'  column. 

Do  this  problem  by  the  computers'  method. 


Exercise 

42 

Fin 

d  the  difference : 

1. 

26.437                     2. 

94.568 

3.    102.4951 

15.254 

29.783 

58.2876 

From: 

4. 

75.093  take  34.267. 

5. 

6.4297  take  3.5824 

6. 

41.7453  take  27.937. 

7. 

3.1111  take  1.4682 

8.  3.1416  take  .9885. 

9.  A  car  containing  24.875  T.  of  coal  was  divided  between 
A  and  B.     A  received  11.375  T.     What  did  B  get  ? 

10.    Show  by   subtracting   6  times   that    a    field    containing 
52.584  A.  can  be  divided  into  6  fields,  each  containing  8.764  A. 


62  ARITHMETIC 

11.  The  highest  rate  of  taxation  paid  in  Cook  County,  Illinois, 
for  the  year  1899  was  $  10.08  on  $  100  property,  and  the  lowest 
$  1.96.     Find  the  difference. 

12.  The  taxes  on  $100  property  for  the  year  1899,  for  the 
different  towns  in  the  city  of  Chicago,  were  as  follows : 

North  Chicago $  6.635 

South  Chicago 6.34 

West  Chicago 7.232 

Hyde  Park 6.35 

Lake 6.426 

Lake  View 6.376 

Jefferson 6,105 

Find  the  difference  between  the  tax  on  $  100  property  in  West 
Chicago  and  in  the  other  towns  of  Chicago. 

Miscellaneous  Exercise  43 

1.  Draw  a  line.  Mark  off  on  this  line  6  parts,  each  J  ft.  long. 
How  many  feet  long  is  the  line  ?  6  times  |  ft.  =  ?  ^  ft.  multi- 
plied by  6  =  ?     I  of  6  ft.  =  ? 

2.  -^  of  12  ft.  =  ?  I  of  $  20  =  ?  30  X  I  =  ? 

12  X  i  ft.  =  ?  20  X  $  f  =  ?  240  X  f  =  ? 

I  of  6  lb.  =  ?  $  I  X  20  =  ?  155  X  f  =  ? 

6  X  I  lb.  =  ?  56  X  f  yd.  =  ?  147  x  f  =  ? 

I  lb.  X  6  =  ?  40  X  I  hr.  =  ?  365  x  §  =  ? 

3.  Find  the  weight  of  12  packages  of  tea  each  containing  |  lb. 
Of  16  yd.  of  carpet  at  ^  lb.  a  yard. 

•  4.  Find  the  weight  of  6  spoons  at  J  oz.  each ;  8  at  J  oz.  each ; 
J  doz.  at  I  oz.  each ;  1 J  doz.  at  |  oz.  each. 

5.  Find  the  weight  of  48  yd.  of  carpet  at  -J-f  lb.  a  yard. 

6.  Find  the  cost  of : 

12  bu.  potatoes  at  $  J  a  bushel. 
6  crates  berries  at  $  |  a  crate. 
10  bu.  apples  at  $  |^  a  bushel. 
6  lb.  coffee  at  f  ^  a  pound. 


SUBTRACTION 


63 


7.  By  what  number   do  you  multiply  to  reduce  gallons   to 
quarts  ? 

Eeduce  to  quarts : 
1  gal.  2  qt.  6  gal.  2  qt.  26  gal.  3  qt. 

3  gal.  1  qt.  12  gal.  1  qt.  35  gal.  2  qt. 

8.  I  bought  3  gal.  2  qt.  of  milk  at  6^  a  quart,  and  gave  a 
dollar  bill  in  payment.     What  change  should  I  get  back  ? 

9.  By  what  number  do  you  multiply  to  reduce  yards  to  feet? 
Reduce  to  feet : 

6  yd.  2  ft.  12  yd.  1  ft.  84  yd.  1  ft. 

8  yd.  1  ft.  18  yd.  2  ft.  72  yd.  2  ft. 

9  yd.  2  ft.  24  yd.  2  ft.  65  yd.  1  ft. 

10.  Find  the  weight  of  an  iron  rod  8  yd.  2  ft.  long  if  1  ft. 
weighs  I  lb. 

11.  What  part  of  f  8  is  |?  2  ?  What  per  cent?  What  part 
of  24  lb.  is  8  lb.  ?  Wliat  per  cent  ?  What  per  cent  of  $  18  is 
$3?     Of  $12  is  ^8?     Of  $36  is  $27?     Of  $  15  is  $  9  ? 

12.  A  grocer  bought  coffee  at  30  ^  a  pound,  and  sold  it  at  a  gain 
of  6  ^  a  pound.  The  gain  is  what  part  of  the  cost  ?  What  per 
cent  of  the  cost  ? 

13.  Find  the  gain  per  cent : 


Cost 

Gain 

Cost 

Gain 

$5 

$1 

75  <f 

25  J^ 

$15 

$5 

42  f 

6f 

$30 

$5 

40^ 

25^ 

$24 

$3 

6if 

4f 

$75 

$50 

90  j^ 

Qf 

$48 

$18 

S6f 

27^ 

$  30 

$4 

27^ 

18^- 

$36 

$3 

72^ 

12;* 

14.    A  merchant  bought  carpet  at  64 ; 
i  P  a  yard.     Find  his  gain  on  each  yard. 


a  yard,  and  sold  it  for 
Find  his  gain  per  cent. 


64 


ARITHMETIC 


15.  A  merchant  bought  handkerchiefs  at  25^  each,  and  sold 
them  for  35  ^.     Find  his  gain  per  cent. 

16.  Find  the  gain  per  cent : 


Cost 

Selling  Price 

Cost 

Selling  Price 

$18 

$24 

$72 

$120 

$44 

$66 

$54 

$60 

$60 

$96 

$82 

$52 

$66 

$77 

$27 

$45 

$63 

$91 

$66 

$72 

$64 

$68 

$48 

$54 

846,361.07 


17.  The  following  is  the  statement  of  the  total  cost  and  the 
net  returns  of  a  peach  orchard  in  Maryland  for  a  period  of  ten 
years : 

Net  returns  for  peach  sales 

Land $2100.00 

Trees 135.00 

Planting,  etc 60.78 

Machinery 170.07 

Cultivation 1320.00 

Fertilizers 769.00 

Taxes 312.06 

Interest 2620.90 

Incidentals 11.88 


Find  the  total  cost  and  the  profits. 

18.    For  the  year  ending  June  30,  1899,  the  exports  from  the 
United  States  to  Great  Britain  and  her  colonies  were : 


To  the  United  Kingdom 
British  North  America 
British  West  Indies 
British  Guiana     . 
British  East  Indies 
British  Asia 
Britisli  Australia 
British  Africa 

Find  their  total  value 


$511,816,476 

89,573,609 

8,751,817 

1,749,545 

4,341,936 

7,732,525 

19,777,129 

16,166,610 


SUBTRACTION  63 

19.  The  total  exports  from  the  United  States  to  all  countries 
for  the  same  year  as  in  the  preceding  example  were  $  1,201,931,222. 
Find  the  value  of  the  exports  to  countries  other  than  Great  Britain 
and  her  colonies. 

20.  How  much  less  was  this  than  the  value  of  the  exports  to 
Great  Britain  and  her  colonies  ? 


CHAPTER  VI 

MULTIPLICATION 

36.  (1)  Beginning  with  $2,  add  by  $2,  till  you  reach  $26. 
What  are  the  $  2  called  ?     Addends.     What  the  result  ?     Sum. 

(2)  In  getting  this  sum  have  you  definitely  thought  of  how 
many  $  2  there  are  ?  Ab.  Do  you  know  from  the  sum  hoiv  many 
there  are  ?     Ao. 

(3)  If  you  add  $  2  to  $  2,  etc.,  till  you  reach  the  sum,  %  182,  do 
you  know  how  many  twos  there  are  ?     No. 

(4)  How  do  you  look  upon  the  sum  $  26  (say)  and  the  $  2  ? 
The  $  26  is  simply  the  sum  of  an  unknown  number  of  $  2. 

(5)  Now  count  the  number  of  $2.  There  are  13.  Did  you 
think  of  this  13  in  the  addition  process  ?     No. 

(6)  Now  consider  this  13  in  relation  to  the  addend  $  2,  and  the 
sum  $  26,  what  new  idea  is  introduced  ?  The  idea  of  how  many 
times  $2  is  repeated  to  make  f  26? 

(7)  Then  what  is  the  number  which  measures  ^2(S?  13.  What 
is  the  unit  of  measure  ?  $  2.  From  what  you  know  of  number, 
say  what  ratio  13  is  ?  The  ratio  of  $  26  to  f  2.  We  say  at  once 
(without  adding)  that  13  times  $  2  is  $  26. 

(8)  In  this  do  we  depend  at  all  on  addition  ?  Yes.  We  first 
find  the  sum,  and  connect  this  in  memory  with  the  number  of 
times  the  addend  is  repeated. 

(9)  But  is  it  then  correct  to  say  that  the  processes  $  2  +  $  2  + 
$  2  ...  =  $  26,  is  identical  with  the  process  13  x  $  2  =  $  26  ?  No; 
for  13  represents  the  "  new  idea  "  referred  to,  and  $  2  has  become 
a  definite  unit  of  measui'e,  which,  with  13  denotes  the  quantity  $26. 
The  addend  has  become  a  factor,  and  the  sum  a  product. 

66 


MULTIPLICATION  67 

37.  Find  the  cost  of  9  yd.  of  cloth  at  1 5  a  yard. 

(1)  Here  we  think  of  f  5  as  a  derived  unit  measuring  the  value  of  1  yd. 
Hence  the  cost  of  9  yd.  is  equal  to  9  x  $5,  or  to  $45. 

(2)  Thus  45,  the  number  of  primary  units  in  the  total  cost,  is  called  the 
product  of  the  number  of  primary  units  in  the  derived  unit  $5,  w^hich  is  5, 
by  the  number  of  units,  viz.  9,  in  the  given  quantity  of  cloth. 

(3)  In  the  above  example  the  total  cost  was  given  by  9  units  of  $  5  each, 
and  after  multiplication  by  45  units  of  $  1  each.  Thus  multiplication  does 
not  change  the  total  cost  {i.e.  the  measured  quantity)  ;  it  changes  only  the 
mtmber  which  measures  it  (in  this  case  from  9  to  45)  by  changing  the  unit 
of  measure,  $5,  to  the  primary  unit,  $1. 

The  numbers  to  be  multiplied  together,  viz.  9  and  5,  are  called  factors  of 
the  product,  i.  e.  of  the  number  that  measures  the  quantity. 

38.  Multiplication  is  the  operation  of  finding  the  number 
of  primary  units  in  a  quantity  expressed  by  a  given  number 
of  derived  units,  or,  more  briefly. 

Multiplication  is  the  operation  of  finding  the  product  of 
two  numbers. 

The  Multiplicand  is  the  derived  unit  of  measure. 

The  Multiplier  denotes  how  many  times  this  unit  of  meas- 
ure is  to  be  repeated,  i.e.  it  denotes  the  ratio  of  the  measured 
quantity  to  the  unit  of  measure. 

39.  8x16  is  read  8  times  1 6. 

$6  X  8  is  read  1 6  multiplied  by  8. 
X  is  the  Sign  of  Multiplication. 

(1)  Require  the  multiplication  table  to  be  memorized  in 
regular  order ;  also,  so  that  it  can  he  given  by  the  pupil  in 
irregular  order,  thus : 

9  X  4  =  36,  9  X  6  =  54,  9  x  10  =  90,  etc. 

(2)  Drill,  requiring  instantaneous  oral  and  written  answers 
to  such  questions  as  :    What  is  6x7?     9x8?     8x9? 

(3)  Drill,  requiring  instantaneous  answers  :   What  is 
6x7  +  4?  9x8  +  3?        .     8x9  +  7? 


68 


ARITHMETIC 


Multiplication  Table 


Twice 

Three  Times 

Four  Times 

Five  Times 

Six  Times 

Seven  Times 

1  is  2 

1  is  3 

1  is  4 

1  is  5 

1  is  6 

1  is  7 

2  "  4 

2  "  6 

2  "  8 

2  "  10 

2  "  12 

2  "  14 

3  "  6 

3  "  9 

3  "  12 

3  "  15 

3  "  18 

3  "  21 

4  "  8 

4  "  12 

4  "  16 

4  "  20 

4  '«  24 

4  "  28 

5  "  10 

5  "  15 

5  "  20 

5  "  25 

5  "  30 

5  "  35 

6  "  12 

6  "  18 

6  "  24 

6  "  30 

6  "  36 

6  ",  42 

7  "  14 

7  "  21 

7  ''  28 

7  "  35 

7  "  42 

7  "  49 

8  "  16 

8  "  24 

8  "  32 

8  "  40 

8  "  48 

8  "  56 

9  "  18 

9  "  27 

9  "  36 

9  "  45 

9  "  54 

9  "  63 

10  "  20 

10  "  30 

10  "  40 

10  "  50 

10  "  60 

10  "  70 

11  "  22 

11  "  33 

11  "  44 

11  "  55 

11  "  66 

11  "  77 

12  "  24 

12  "  36 

12  "  48 

12  "  60 

12  "  72  • 

12  "  84 

Eight  Times 

Nine  Times 

Ten  Times 

Eleven  Times 

Twelve  Times 

1  is  8 

1  is   9 

1  is  10 

1  is  11 

1  is   12 

2  "  16 

2  "   18 

2  ' 

20 

2  "  22 

2 

'  24 

3  "  24 

3  "  27 

3  ' 

30 

3  "  33 

3 

'  36 

4  ''  32 

4  "  36 

4  ' 

40 

4  "  44 

4 

'  48 

5  "  40 

5  "  45 

5  ' 

50 

5  "  55 

5 

'  60 

6  "  48 

6  "  54 

6  ' 

60 

6  "  66 

6 

'  72 

7  "  56 

7  "  63 

7  ' 

70 

7  "  77 

7 

'  84 

8  "  04 

8  "  72 

8  ' 

80 

8  "  88 

8 

'  96 

9  •'  72 

9  "  81 

9  ' 

90 

9  "  99 

9 

'  108 

10  "  80 

10  "  90 

10  ' 

100 

10  "  110 

10 

'  120 

11  "  88 

11  "  99 

11  ' 

110 

11  "  121 

11 

'  132 

12  "  96 

12  "  108 

12  ' 

120 

12  "  132 

12 

'  144 

MULTIPLICATION  69 

40.  In  the  preceding  diagram  we  have  36  dots,  signifying 
36  units  of  any  kind,  arranged  in  4  rows  of  9  dots  each,  and 
at  the  same  time  9  rows  of  4  dots  each.  Hence  we  think  of 
36  as  equal  to  4  times  9  or  9  times  4.  Arrange  the  dots  to 
show  that  36  is  equal  to  3  x  12  or  12  x  3,  and  also  to  2  x  18 
or  18  X  2. 

4  and  9  are  called  factors  of  36,  and  36  is  called  the  product 
of  4  and  9. 

This  illustrates  the  law  of  commutation,  a  law  of  great 
importance  in  Arithmetic. 

Thus  we  think  of  24  as  equal  to  2  x  12  or  12  x  2,  3  x  8  or 
8  X  3,  4  X  6  or  6  X  4. 


41.  If  in  the  above  arrangement  we  think  of  each  dot  as 
representing  $  1,  then  the  diagram  shows  that  i  12  -j-  i2  =  6. 

What  other  measurement  is  shown  by  the  same  arrange- 
ment ? 

Exercise  44 

1.  Arrange  dots,  representing  any  units,  to  show  that 

18  =  3  X  6,  or  6  X  3  =  2  X  9,  or  9  X  2. 

2.  Give  the  factors  of  45  (9  x  5  or  5  x  9),  66,  56,  72,  96,  63, 
90,  54,  99,  84,  132,  108. 

3.  Give  the  factors  of  9,  16,  25,  36,  49,  64,  81,  100,  121,  144. 

4.  Arrange  30  dots  to  show  how  often  30  is  measured  by  10. 
By  3. 

5.  How  often  is  72  measured  by  9  ?  8  ?  6  ?  12  ?  4?  18  ?  3? 
24?  2?  36? 

6.  If  one  factor  of  96  is  12,  what  is  the  other?     If  one  factor 
is  8,  what  is  the  other  ? 

7.  What  will  5  yd.  of  cloth  cost  at  $4  a  yard?     What  will 
4  yd.  cost  a  $  5  a  yard  ? 


70  ARITHMETIC 

8.  If  9  men  can  do  a  piece  of  work  in  6  da.,  how  long  will  it 
take  1  man  to  do  it  ?  If  6  men  can  do  a  piece  of  work  in  9  da., 
how  long  will  it  take  1  man  to  do  it  ? 


Scale:  i  in.  =  1  in. 

42.  (1)  Find  the  area  of  an  oblong  5  in.  long  and  3  in. 
wide. 

Let  the  oblong  be  divided  into  3  strips  by  lines  1  in.  apart,  as  in  the  figure. 
The  area  of  1  strip  =  5  sq.  in. 
. '.  the  area  of  the  oblong  =  3  x  5  sq.  in. 
=  15  sq.  in. 

Prove  the  answer  correct  by  dividing  the  oblong  into  square  inches,  and 
counting. 

Make  a  drawing  to  show  that  the  area  is  equal  to  6  x  3  sq.  ft. 
Thus  we  may  think  of  the  area  as  either  5  x  3  or  3  x  5  sq.  in. 

(2)    Reduce  9  pk.  6  qt.  to  quarts. 

9  pk.  =  9  X  8  qt.  =  72  qt. 
.  •.  9  pk.  6  qt.  =  78  qt. 
Here  the  problem  is  to  add  6  qt.  to  9  units  of  8  qt.  each. 

Note.  —  By  the  law  of  commutation  we  may  think  of  9  pk.  as  9  x  8  qt., 
or  as  8  X  9  qt.  Thus  in  reducing  pecks  to  quarts  we  may  use  8  as  a  constant 
nuiltiplicr. 

What  constant  multiplier  will  reduce  yards  to  feet  ?  Feet  to  inches  ? 
Gallons  to  quarts  ?     Quarts  to  pints  ?    Weeks  to  days  ? 


MULTIPLICATIO:Nr 


71 


Scale:  ^  in.  =  1  in. 

(3)    Find  the  volume  of  a  rectangular  solid  5  in.  long, 
4  in.  wide,  and  3  in.  thick. 

Let  the  solid  be  divided  into  3  slices  by  horizontal  planes  1  in.  apart. 
Let  the  upper  slice  be  divided  into  5  rovv^s  by  vertical  planes  1  in.  apart. 
Let  the  right-hand  row  be  divided  into  4  cu.  in.  by  vertical  planes  1  in. 
apart. 

The  volume  of  1  row  =  4  cu.  in. 

The  volume  of  5  rows  or  1  slice  =  5  x  4  cu.  in. 
The  volume  of  3  slices  or  the  solid  =  3  x  5  x  4  cu.  in. 

=  60  cu.  in. 


Exercise  45 

1.  Make  a  drawing  to  show  that  an  oblong  6  in.  long  and  4  in. 
wide  contains  24  sq.  in. 

2.  Find  the  area  of  the  following  oblongs  :  6  in.  by  8  in. ;  7  in. 
by  9  in. ;  8  in.  by  11  in. ;  9  in.  by  12  in. 

3.  How  do  you  find  the  number  of  units  of  area  in  an  oblong  ? 

4.  Find  the  area  of  the  floor  of  each  of  the  rooms  whose 
dimensions  are  6  yd.,  5  yd. ;  8  yd.,  8  yd. ;  12  yd.,  10  yd. ;  12  yd., 
12  yd. 


72  ARITHMETIC 

5.  A  garden  bed  5  ft.  long  and  3  ft.  wide  is  surrounded  by  a 
walk  1  ft.  wide.  Make  a  drawing  to  show  the  entire  area  of  the 
bed  and  walk.     What  is  this  area  ?     (Scale  1  in.  to  1  ft.) 

6.  Find  the  number  of  square  inches  in  each  of  the  stamped 
linen  squares  whose  sides  are,  respectively,  6,  8,  10,  and  12  in. 

7.  Find  the  number  of  square  feet  in  a  square  yard ;  of  square 
inches  in  a  square  foot. 

8.  A  rug  occupies  the  centre  of  a  square  room,  and  is  every- 
where 2  ft.  from  the  wall.  If  the  room  is  14  ft.  long,  find  the 
size  of  the  rug. 

9.  How  long  is  a  township  ?     How  wide  ?     What  is  its  area? 

10.  Make  a  drawing  to  show  the  volume  of  a  rectangular  solid 
2  in.  by  3  in.  by  4  in.  What  is  it  ?  How  do  you  find  the  volume 
of  a  rectangular  solid  ? 

11.  Find  the  volume  of  each  of  the  rectangular  solids  whose 
dimensions  are:    2  in.,  3  in.,  5  in. ;   4  in.,  6  in.,  9  in. ;    4  in.,  5  in., 

7  in. ;  6  in.,  6  in.,  6  in. ;  2  ft.,  2  ft.,  2  f t. ;  3  units  of  length,  4  units, 
5  units.  '' 

12.  Find  the  number  of  cubic  feet  in  a  cubic  yard.  Of  cubic 
inches  in  a  cubic  foot. 

13.  Make  a  drawing  to  show  the  perimeter  of  an  oblong  3  in. 
by  5  in.  What  is  it?  How  do  you  find  the  perimeter  of  an 
oblong  ? 

14.  Find  the  perimeter  of  rooms  whose  dimensions  are :  6  yd., 

8  yd. ;  7  yd.,  9  yd. ;  8  yd.,  11  yd. ;  11  ft.,  12  ft. ;  12  ft.,  15  ft. 

15.  Find  the  number  of  yards  of  braid  needed  to  carpet  a  rug 
4  yd.  long  and  3  yd.  wide.     What  does  it  cost  at  8^  a  yard  ? 

16.  How  long  will  it  take  a  bicyclist  to  ride  around  a  square 
township  at  8  mi.  an  hour  ? 

Exercise  46 
1.    Reduce  to  lower  denominations  : 
(1)  5  yd.  2  ft. ;  7  yd.  1  ft. ;  8  yd.  2  ft. ;  9  yd.  1  ft. ;  11  yd.  1  ft.; 
12  yd.  2  ft. 


MULTIPLICATION"  73 

(2)  5  ft.  4  in. ;  8  ft.  10  in.;  7  ft.  8  in.;  11  ft.  3  in. ;  9  ft.  7  in.; 
12  ft.  6  in. 

2.  Find  the  cost  of  making  a  sidewalk  10  yd.  2  ft.  long  at 
25^  a  foot. 

3.  Find  the  cost  of  5  curtains,  each  3  yd.  1  ft.  long,  at  6^ 
a  foot. 

Keduce  to  lower  denominations  : 

4.  3  sq.  yd.  5  sq.  ft. ;  5  sq.  yd.  7  sq.  ft. ;  12  sq.  yd.  6  sq.  ft. ; 

8  sq.  yd.  8  sq.  ft. ;  9  sq.  yd.  2  sq.  ft. ;  11  sq.  yd.  10  sq.  ft. 

5.  6  qt.  1  pt.;  8  qt.  1  pt.;  11  qt.  1  pt.;  7  pk.  6  qt.;  9  pk.  4  qt.; 

9  bu.  3  pk.;  8  bu.  2  pk.;  11  bu.  3  pk.;  7  gal.  2  qt.;  9  gal.  3  qt. 

6.  Find  the  cost  of  3  gal.  2  qt.  of  milk  at  6^  a  quart. 

7.  A  grocer  sold  6  bu.  3  pk.  of  potatoes  at  18^  a  peck.  Find 
the  selling  price. 

Reduce  to  lower  denominations: 

8.  7  wk.  4  da.;  9  wk.  2  da.;  11  wk.  6  da.;  8  wk.  1  da.;  12  wk. 
5  da.;  5  hr,  40  min.;  8  hr.  9  min. ;  9  hr.  22  min. ;  12  hr.  45  min.; 
5  da.  4hr.;  8  da.  2  hr. 

9.  How  many  hours  are  there  in  one  week  ? 

43.   What  is  the  cost  of  6  town  lots  at  $  894  a  lot  ? 

Here  we  think  of  the  whole  cost  as  6  units  of  $  894  each. 
Explanation.  —  The  unit  $  894  may  be  considered  as  made  up  of  4  units 
of  one  dollar,  9  units  of  ten  dollars,  and  8  units  of  one  hundred  dollars. 

6x4  units  of  one  dollar  =  24  units  of  one  dollar  =  2  units  of 
*  ^^4       tgn  dollars  +  4  units  of  one  dollar. 

6  6x9  units  of  ten  dollars  =  54  units  of  ten  dollars. 


S  5364  ^^  units  of  ten  dollars  +  2  units  of  ten  dollars  =  56  units  of  ten 

dollars  =  5  units  of  one  hundred  dollars  +  6  units  of  ten  dollars. 

6x8  units  of  one  hundred  dollars  =  48  units  of  one  hundred  dollars. 

48  units  of  one  hundred  dollars  +  5  units  of  one  hundred  dollars  =  53 
units  of  one  hundred  dollars  =  5  units  of  one  thousand  dollars  +  3  units  of 
one  hundred  dollars. 

5  units  of  one  thousand  dollars  +  3  units  of  one  hundred  dollars  +  6  units 
of  ten  dollars  +  4  units  of  one  dollar  =  $  5364. 


74  ARITHMETIC 

Exercise  47 
Multiply  separately : 

1.  231  by  2,  4,  6,  8,  10,  and  12. 

2.  690  by  3,  5,  7,  9,  and  11. 

3.  897  by  4,  6,  8,  10,  and  12. 

4.  2463  by  3,  5,  7,  9,  and  11. 

5.  5781  by  2,  4,  8,  10,  and  12. 

6.  9654  by  3,  5,  7,  9,  and  11. 

7.  8267  by  2,  4,  6,  8,  10,  and  12. 

8.  5280  by  3,  5,  7,  9,  and  12. 

9.  1728  by  2,  4,  6,  8,  10,  and  12. 

10.  4840  by  3,  5,  7,  9,  and  11. 

11.  63,360  by  2,  4,  6,  8,  10,  and  12. 

12.  24,793  by  3,  6,  7,  9,  and  11. 

13.  98,654  by  2,  4,  6,  8,  10,  and  12. 

14.  89,743  by  3,  5,  7,  9,  and  11. 

15.  64,789  by  2,  4,  5,  8,  10,  and  12. 

Exercise  48 

1.  How  do  you  find  the  perimeter  of  a  room  ?     The  area  of  an 

oblong  ? 

2.  Find  the  perimeter  of  a  room  18  ft.  long  and  12  ft.  wide. 
What  is  the  area  of  an  oblong  whose  length  is  equal  to  the  perim- 
eter of  this  room  and  whose  width  is  9  ft.  ?  (Draw  this  oblong 
on  the  blackboard,  scale  =  1  in.  to  1  ft.) 

3.  The  length  of  a  room  is  15  ft.  and  the  width  10  ft.  Find  its 
perimeter.  Find  the  area  of  an  oblong  whose  length  is  equal  to 
the  perimeter  of  this  room  and  whose  width  is  8  ft. 

4.  What  is  the  perimeter  of  a  room  whose  dimensions  are 
15  ft.  and  10  ft.?  What  is  the  area  of  the  four  walls  of  a  room 
whose  dimensions  are  15  ft.  and  10  ft.,  and  height  8  ft.  ? 


MULTIPLICATION  75 

5.    (a)  Find  the  perimeters  of  rooms  whose  dimensions  are: 


Length 

Width 

Height 

16  ft. 

14  ft. 

8  ft. 

17  ft. 

15  ft. 

8  ft. 

20  ft. 

18  ft. 

9  ft. 

22  ft. 

20  ft. 

12  ft. 

(6)  Find  the  areas  of  the  four  walls  of  these  rooms. 

6.  A  cord  of  wood  is  8  ft.  long,  4  ft.  wide,  4  ft.  high.  How 
many  cubic  feet  does  it  contain  ? 

7.  Find  the  number  of  cubic  inches  in  a  cubic  foot. 

8.  A  gallon  of  water  will  exactly  fill  a  rectangular  box  11  in. 
long,  -7  in.  wide,  and  3  in.  high.  Find  the  number  of  cubic  inches 
in  a  gallon. 

9.  There  are  1760  yd.  in  1  mi.  Find  the  number  of  yards  in 
2  mi.;  5  mi.;  8  mi.;  9  mi.;  12  mi. 

10.  There  are  5280  ft.  in  1  mi.     Find  the  number  of  feet  in 
3mi. ;  6  mi.;  7  mi.;  9  mi.;  11  mi. 

11.  Reduce  to  square  inches:  5  sq.  ft.;  8  sq.  ft.;  10  sq.  ft.; 
12  sq.  ft. 

12.  Reduce  to  square  feet:  3547  sq.  yd.;  8426  sq.  yd.;  9819 
sq.  yd. 

13.  One  square  mile  contains  640  A.     Find  how  many  acres 
there  are  in  6  sq.  mi. ;  8  sq.  mi. ;  10  sq.  mi. ;  12  sq.  mi. 

14.  Reduce  to  cubic   inches:   4  cu.   ft.;   5  cu.   ft.;   7  cu.   ft.; 
9  cu.  ft.;  11  cu.  ft. 

15.  How  do  you  reduce  weeks  to  days  ?  gallons  to  quarts  ? 

16.  Reduce  to  days:  453  wk.;  769  wk.;  827  wk.;  852  wk. 

17.  Reduce  to  quarts:  765  gal. ;  917  gal. ;  763  gal.;  789  gal. 

18.  Reduce  to  quarts:  735  pk.;  892  pk.;  679  pk. ;  728  pk. 

19.  Reduce  to  days:  3  yr.;  5  yr.;  6  yr.;  8  yr.;  11  yr.;  12  yr. 
(lyr.  =  365da.) 


76  ARITHMETIC 

44.   What  is  the  cost  of  76  town  lots  at  f  894  a  lot  ? 

'f  894  The  explanation  is  similar  to  that  given  in  §  43.    Since, 

nn         when  7  is  used  as  a  multiplier,  the  4  units  of  one  dollar  are 

— — -        multiplied  by  7  tens,  the  product  is  the  same  as  that  found 

0004         y^y  multiplying  4  units  of  te?i  dollars  by  7.     This  is  28  units 

6258  of  ten  dollars,  and  is  equal  to  2  units  of  one  hundred  dollars 

$67944         ^^^  ^  units  of  ten  dollars.     Hence  the  8  is  written  under 

the  6  in  the  tens'  column,  and  the  2  is  carried  to  be  added 

in  the  hundreds'  column,  and  so  on. 

To  prove  the  answer  correct,  multiply  76  by  894  ;  thus : 

76 
894 
304 
684 


.♦.  the  answer  is  correct. 

67944 

Exercise 

49 

Multiply  and  prove  your  answers 

correct : 

1.   423  by  36. 

11. 

8647  by    365. 

2.   479  by  32. 

12. 

7245  by    168. 

3.   295  by  16. 

13. 

8939  by    224. 

4.   798  by  24. 

14. 

6558  by    144. 

5.   581  by  52. 

15. 

9275  by    231. 

6.   649  by  27. 

16. 

9475  by  1760. 

7.   959  by  24. 

17. 

8213  by  5280. 

8.   764  by  31. 

18. 

4781  by  1728. 

9.   953  by  56. 

19. 

5893  by  2240. 

10.   825  by  48. 

20. 

6439  by  1728. 

Multiply : 

Exercise 

50 

1.   8245  by  684. 

6. 

8746  by  675. 

2.   7639  by  797. 

7. 

9687  by  897. 

3.   5927  by  395. 

8. 

4786  by  478. 

4.   4399  by  927. 

9. 

9467  by  769. 

5.   8999  by  868. 

10. 

8769  by  567. 

MULTIPLICATION  77 

45.  (1)  A  drover  bought  36  horses  at  $145  a  head,  and  96 
cows  at  128  a  head.     Which  cost  the  more,  and  how  much? 

Here  the  problem  is  to  find  the  difference  between  36  units  of  $  145  each 
and  96  units  of  $28  each.  Multiply  $  145  by  36  and  $28  by  96,  and  find  the 
difference. 

(2)  A's  barn  cost  il75,  his  house  16  times  as  much,  and 
his  farm  cost  as  much  as  both.     What  was  the  cost  of  all  ? 

Here  the  problem  is  to  find  the  cost  of  1  +  16  +  17,  or  34  units  of  $  175 
each.     Multiply  $  175  by  34  and  find  the  cost  of  all. 

Exercise  51 

1.  A  dry  goods  firm  had  28  clerks,  and  paid  an  average  salary 
of  $45  a  month.    Find  the  total  amount  of  their  monthly  salaries. 

2.  A  real  estate  dealer  bought  8  lots,  and  sold  them  at  a  gain 
of  $  250  on  each  lot.     Find  his  total  gain. 

3.  A  dealer  bought  150  head  of  cattle  and  47  mules.  He  made 
a  profit  of  f  13  a  head  on  the  former  and  $  17  each  on  the  latter. 
Find  his  total  gain. 

4.  A  vessel  took  3  da.  to  make  a  trip.  If  the  average  rate 
was  12  mi.  an  hour,  find  the  length  of  the  trip. 

5.  A  boy  left  home  on  his  bicycle,  and  rode  5  hr.  at  the  rate 
of  8  mi.  an  hour.  He  returned  home  at  the  rate  of  6  mi.  an  hour. 
How  far  was  he  from  home  4  hr.  after  starting  back  ?  Draw  a 
line  and  mark  off  the  distances. 

6.  A  train  started  from  St.  Louis  and  travelled  12  hr.  at  the 
rate  of  35  mi.  an  hour.  The  next  day  it  returned  at  the  rate  of 
32  mi.  an  hour.  Find  its  distance  from  St.  Louis  3  hr.  after  start- 
ing back. 

7.  A  farmer  raised  a  crop  of  58  bu.  of  corn  an  acre  from  48  A. 
Find  the  number  of  bushels  of  corn.  What  is  its  value  at  38  ^  a 
bushel  ? 

8.  A  farmer  raised  a  crop  of  37  bu.  of  oats  an  acre  from  65  A. 
Find  the  value  of  the  oats  at  26  ^  a  bushel. 


78  ARITHMETIC 

9.   Make  out  a  bill  for  the  following  goods : 

23  yd.  cotton  @  11  f^ ;  13  yd.  gingham  @  23^ ; 
25  yd.  flannel  @  37^;  18  yd.  tweed  (a)  $1.50; 
12  yd.  serge  @  $1.75;   6  yd.  broadcloth  @  $4.50. 

10.  A  produce  merchant  exchanged  48  bii.  of  oats  at  39^  per 
bushel  and  13  bbl.  of  apples  at  $3.85  a  barrel  for  200  lb.  of  butter 
at  37  ^  a  pound.     How  much  should  he  pay  to  settle  the  account  ? 

11.  A  grain  dealer  buys  4795  bu.  of  wheat  in  Chicago  at  63  ^  a 
bushel,  and  ships  it  to  New  York  at  a  cost  of  3^  a  bushel.  Find 
his  gain  if  he  sells  it  in  New  York  for  71^  a  bushel. 

12.  A  man  bought  51  horses  at  $97  each,  and  sold  them  at 
$  136  each.     How  much  did  he  gain  ? 

13.  Find  the  amount  of  the  following  bill: 

63  brooms,  at  16  ^  each ;  ' 

13  yd.  print,  at  11  ^  per  yard ; 
17  lb.  tea,  at  35  ^  per  pound ; 
4  doz.  oranges,  at  4  ^  each ; 
287  lb.  sugar,  at  5^  per  pound; 
84  eggs,  at  13^  per  dozen. 

14.  A  fruit  dealer  sold  apples  at  the  rate  of  6  for  5^.  What 
was  the  price  per  dozen  ?  If  he  sold  oranges  at  the  rate  of  3  for 
5^,  what  was  the  price  per  dozen? 

15.  A  fruit  dealer  paid  8^  a  dozen  for  bananas,  and  sold  them 
at  the  rate  of  4  for  5^.  Find  his  gain  on  a  bunch  containing 
6  doz. 

16.  Bought  oranges  at  the  rate  of  18  ^  a  dozen,  and  sold  them 
at  the  rate  of  6  oranges  for  15^.  How  much  did  I  gain  on  11 
boxes,  each  containing  20  doz.  ? 

17.  If  in  the  previous  example  two  boxes  were  spoiled,  what 
was  the  gain  ? 

18.  How  far  will  a  bicyclist  ride  in  12  da.,  if  he  rides  6  hr.  a 
day  at  8  mi.  an  hour  ? 


MULTIPLICATION 


79 


19.  Two  vessels  start  from  the  same  point  and  travel,  the  one 
down  a  river  at  the  rate  of  15  mi.  an  hour,  the  other  up  the  river 
at  the  rate  of  9  mi.  an  hour.     How  far  will  they  be  apart  in  8  hr.  ? 

20.  If  the  first  vessel  travelled  np  the  river  at  the  rate  of  12  mi. 
an  hour,  how  far  apart  w^ould  they  be  in  8  hr.  ? 

21.  A  speculator  bought  45  A.  of  land  at  $  65  an  acre,  and 
63  A.  at  $  78  an  acre.  If  he  sold  the  whole  at  $  75  an  acre,  how 
much  did  he  gain  or  lose  ? 

22.  In  1898  there  were  shipped  from  the  melon  district  of 
Indiana  1156  carloads  of  melons.  If  the  average  cost  of  ship- 
ping was  $  29.75  a  car,  find  the  total  cost. 

23.  The  wholesale  price  of  a  dressed  beef  weighing  800  lb.  is 
given  in  the  following  table : 


Pounds 

Per  Pound 

Pounds 

Per  Pound 

Forequarters : 

Hindquarters : 

Roast    .     .     . 

76 

16^  cents 

Round .     .     . 

180 

9    cents 

Plate     .     .     . 

90 

3^  cents 

Loin     .     ,     . 

140 

16^  cents 

Shank  .     .     . 

24 

3    cents 

Suet     .     .     . 

24 

4    cents 

Chuck  .     .     . 

210 

6|  cents 

Flank  .     .     . 

24 

3    cents 

Shank  .     .     . 

32 

2|  cents 

Find  the  cost  of  the  forequarters  and  of  the  hindquarters. 

24.  In  the  previous  example  what  is  the  entire  cost  of  the 
beef,  and  how  much  less  is  this  than  9  ^  a  pound  ? 

25.  The  grain  receipts  of  the  city  of  Chicago,  Aug.  24,  1899, 
were  80  cars  of  wheat,  365  of  corn,  and  480  of  oats.  If  a  car  of 
wheat  contains  400  bu.,  of  corn  400  bu.,  and  of  oats  680  bu.,  find 
the  number  of  bushels  of  each  kind  of  grain  received. 

26.  A  farmer  bought  80  A.  of  land  at  f  25  an  acre,  and  drained 
it  at  a  cost  of  $  475,  thus  increasing  its  value  to  $  45  an  acre. 
Find  the  increase  in  the  value  of  the  farm  above  the  cost  of 
drainage. 


80  ARITHMETIC 

27.  A  section  of  land  in  the  state  of  Missouri,  containing 
108,200  A.,  was  drained  at  a  cost  of  $  700,000,  thus  increasing 
its  vahie  from  $5  to  $40  an  acre.  Find  the  total  increase  in 
value  above  the  cost  of  drainage. 

46.   Multiply  .948  by  6. 

.948 

^         948  thousandths  multiplied  by  6  equals  5688  thousandths  or 

r    6.688. 


5.688 

Exercise  62 

Multiply : 

1.  .5  by  9.  6.  .842  by  9.  11.  4.79  by  32. 

2.  .8  by  3.  7.  .1416  by  25.  12.  3.295  by  16. 

3.  .26  by  3.  8.  .988  by  76.  13.  .7568  by  144. 

4.  .39  by  8.  9.  3.54  by  12.  14.  8.754  by  172. 

5.  .624  by  3.  10.  .543  by  36. 

15.  Show  by  measuring  a  plate  that  its  circumference  is  more 
than  three  times  its  diameter. 

16.  Draw  a  circle  4  in.  in  diameter.  TJie  circumference  of  a 
circle  is  3.1416  ti7nes  the  diameter.  Multiply  3.1416  by  4  and  find 
the  length  of  the  circumference.  Measure  the  circumference  and 
see  if  your  answer  appears  to  be  correct. 

17.  By  what  do  you  multiply  3.1416  to  find  the  circumference 
of  a  circle  ?  Find  the  circumference  of  a  circle  whose  diameter 
is  3  in. 

18.  Find  the  circumferences  of  circles  whose  diameters  are  6  in., 
8  in.,  10  in.,  7  ft.,  9  yd.,  2  mi. 

19.  Find  the  circumference  of  a  circular  flower  bed  whose 
diameter  is  4  ft. 

20.  A  circular  lake  is  5  mi.  in  diameter.  How  many  miles  does 
a  boy  go  in  skating  around  the  lake  ? 

21.  Find  the  volume  of  a  rectangular  piece  of  wood  2  ft.  by 
3  ft.  by  4  ft.  If  this  weighs  36.125  lb.  per  cubic  foot,  find  its 
weight. 


,    MULTIPLICATION  81 

22.  Find  the  weight  of  a  rectangular  solid  of  oak,  3  ft.  by  2  ft. 
by  1  ft.,  weighing  47.375  lb.  per  cubic  foot. 

23.  A  cubic  foot  contains  7.48  gal.  of  water.  Find  how  many 
gallons  of  water  will  till  a  tin  box  containing  8  cu.  ft.  ? 

24.  A  cubic  foot  contains  7.48  gal.  of  water.  Find  how  many 
gallons  can  be  poured  into  a  tin-lined  box,  whose  interior  dimen- 
sions are  3  by  4  by  6  ft. 

25.  A  drover  bought  12  sheep  at  $5,375  per  head,  36  at 
14.625,  and  212  at  $4,125.     Find  the  total  cost. 

26.  Multiply  each  of  the  following  numbers  by  10 :  .43 ;  .576 ; 
4.23;  .017;  89.4263. 

27.  In  multiplying  by  10  how  many  places  to  the  right  do  you 
move  the  decimal  point  ? 

28.  Write  down  the  product  of  each  of  the  following  numbers 
multiplied  by  10 :  7.4 ;  8.946 ;  5.32 ;  .008  ;  62.9347. 

29.  Multiply  each  of  the  following  by  100 :  .435;  8.027;  9.12; 
46.5928. 

30.  In  multiplying  by  100  how  many  places  to  the  right  do 
you  move  the  decimal  point  ? 

31.  Write  down  the  product  of  each  of  the  following  numbers 
multiplied  by  100 :  4.95;  84.793;  9.714;  52.1967. 

Miscellaneous  Exercise  52  (a) 

1.  Show  by  a  drawing  that  li  ft.  =  |  ft.     Show  by  a  drawing 
that  If  ft.  =  I  ft.     Show  by  a  drawing  that  2|  ft.  =  -V-  ft. 

2.  If  ft.  =  I  ft.  21  lb.  =  ?  lb.  3i  =  ? 
2|  ft.  =  1  ft.                  41  lb.  =  ?  lb.                      42  =  ? 


3iyd.  =  iyd.  2Llb.=?lb.  6f 

3.  20  X  f  =  ?  12  X  I  =  ?  14  X  2f 


9 


? 


24  X  I  =  ?  12  X  If  =  ?  40  X  If  =  ? 

16  X  I  =  ?  18  X  If  =  ?  45  X  2|  =  ? 

42  X  J  =  ?  36  X  3J  =  ?  15  X  3|  =  ? 


82 


ARITHMETIC 


4.  What  is  the  cost  of  10  lb.  of  sugar  at  5|^  a  pound  ?  Of 
8  yd.  of  silk  at  $  2i  a  yard  ? 

5.  What  is  the  weight  of  16  yd.  of  carpet  at  2^  lb.  a  yard? 
Of  8  yd.  at  If  lb.  a  yard  ?  Of  8  sq.  yd.  of  oil-cloth  at  If  lb.  per 
square  yard  ?  Of  24  sq.  yd.  of  floor  oil-cloth  at  2|  lb.  per  square 
yard  ? 

6.  Find  the  cost  of : 

40  yd.  carpet  lining  at  2|  ^  a  yard. 
12  yd.  binding  at  f  ^  a  yard. 
100  fish  hooks  at  ^  ^  each. 
3  doz.  penholders  at  2|  ^  each. 

7.  Reduce  to  pecks : 

8  bu. ;  6  bu.  3  pk. ;  9  bu.  2  pk. ;  12  bu.  1  pk. ;  25  bu.  3  pk. 

8.  rind  the  selling  price  of  6  bu.  2  i)k.  of  apples  at  25  ^  a 
peck.     Of  8  bu.  3  pk.  of  potatoes  at  18  )^  a  peck. 

9.  How  many  quarts  in  one  bushel  ?     Eeduce  to  quarts : 

3  bu.  6  qt. ;  12  bu.  18  qt. ;  27  bu.  8  qt. ;  45  bu.  9  qt. ;  33  bu.  6  qt. 

10.  Find  the  selling  price  of  3  bu.  8  qt.  of  beans  at  8  ^  a  quart. 

11.  A  merchant  bought  cloth  at  25^  a  yard  and  sold  it  at  a  loss 
of  10  ^  a  yard.  The  loss  is  what  part  of  the  cost  ?  What  per 
cent  of  the  cost  ? 

12.  Find  the  loss  per  cent : 


Cost 

Loss 

Cost 

Loss 

$   3 

$    1 

75^ 

5J? 

.140 

$16 

5Qf 

21,<* 

182 

$   4 

27^ 

18^ 

$6.3 

$42 

84  J? 

12^ 

13.  A  dealer  paid  $36  apiece  for  bicycles  and  sold  them  for 
$33.  How  much  did  he  lose  on  each  bicycle  ?  What  per  cent  of 
the  cost  ? 


MULTIPLICATION 


83 


14.    Find  the  loss  per  cent : 


Cost 

Selling  Price 

Cost 

Selling  Peice 

$25 

$20 

50^ 

30^ 

$26 

$15 

99^ 

90^ 

$  3 

$   2.25 

45^ 

i2f 

$  6 

$   3.75 

39^ 

26^ 

$  9 

$   7.29 

16  f 

4^ 

15.    Great  Britain's  "grocery  bill"  with  the  United  States  for 
1899  was : 


Canned  beef 2,066,308 

Salted  beef 1,080,351 

Tallow 1,538,114 

Butter 1,705,190 

Cheese 2,063,409 

Petroleum 8,563,518 

Tobacco 7,808,850 

Horses 3,024,952 

Sheep 702,347 


Corn $27,512,398 

Wheat 55,367,397 

Flour 41,335,609 

Fresh  beef 23,456,488 

Live  cattle 28,213,572 

Bacon.    ......  30,312,477 

Hams  .......  16,366,864 

Lard 12,310,730 

Pickled  pork     ....  3,119,067 

Fresh  pork 2,686,191 

Find  the  total. 

16.  In  addition  to  the  above  Great  Britain  purchased  in  the 
United  States  cotton  to  the  value  of  $99,709,352;  find  the  total. 

17.  In  1889  the  total  number  of  vessels  passing  through  the 
Sault  Ste.  Marie  canal  was  5579.  The  total  freight  7,516,022  T. 
In  1899  the  total  number  of  vessels  was  20,055,  the  total  freight 
25,255,810  T.     Find  the  increase  in  each  case. 

18.  The  average  price  of  cotton  was  2^^  a  pound  more  in  1899 
than  in  1898.     Find  the  increase  in  the  price  of  a  bale*  of  500  lb. 

19.  Find  the  increase  in  the  value  of  a  cotton  crop  of  625  bales 
due  to  an  advance  of  IJ^  a  pound. 

20.  Multiply: 

.643  72.95  3.624  6.438 

48  29  88  75 


CHAPTER  VII 

DIVISION 

47.  What  will  4  oranges  cost  at  5  ^  apiece  ?  If  4  oranges 
cost  20  ^,  what  is  the  cost  of  each  ?  What  must  5^  be  multi- 
plied by  to  get  20^? 

At  i  5  a  yard,  how  much  will  9  yd.  of  cloth  cost  ?  What 
must  $  3  be  multiplied  by  to  get  i  27  ?  At  ^  3  a  yard,  how 
many  yards  can  be  bought  for  $21  ? 

48.  Of  what  product  are  5  and  6  the  factors?  (30.)  If  5 
is  one  factor  of  30,  what  is  the  other  ?  Of  what  product  are 
12  and  4  the  factors  ?  If  4  is  one  factor  of  48,  what  is  the 
other  ?  If  6  is  one  factor  of  42,  what  is  the  other  ?  If  9  is 
one  factor  of  63,  what  is  the  other  ?  4  is  one  factor  of  each 
of  the  following  numbers  :  what  are  the  other  factors  ?  24, 
36,  20,  28,  and  16. 

49.  In  Multiplication  we  are  given  two  factors  and  we  are 
required  to  find  their  product. 

In  Division,  on  the  other  hand,  we  are  given  the  product, 
and  also  one  of  the  factors,  and  we  are  required  to  find  the 
other  factpr. 

Thus  :  Find  how  many  yards  of  cloth  at  $5  a  yard  can  be 
bought  for  145? 

In  this  problem,  45,  the  number  measuring  the  cost  of  the 
cloth,  is  the  product  of  two  factors.  One  of  these  is  5,  the 
given  number,  which  measures  the  value  of  the  unit,  and 
the  other  is  9,  which  is  the  number  of  yards. 

84 


DIVISION  85 

If  9  yd.  of  cloth  cost  145,  what  will  1  yd.  cost  ? 
As  before,  45  is  the  product  of  two  factors.     The  given 
factor  is  9,  and  the  required  factor  5.      Therefore   1  yd. 

costs  15.  - 

50.  Division  is  the  operation  of  finding  either  of  two  fac- 
tors, when  their  product  and  the  other  factor  are  given. 

The  factor  found  is  called  the  Quotient.  It  shows  how 
often  the  Divisor  is  contained  in  the  Dividend. 

The  given  factor  is  called  the  Divisor. 

The  given  product  of  the  Quotient  and  Divisor  is  called 
the  Dividend. 

When  the  Divisor  is  not  contained  an  exact  number  of 
times,  the  excess  is  called  the  Remainder.     See  §  57". 

51.  The  sign  of  Division  is  -^ ;  thus  $8-f-f2  =  4  is  read 
$  8  divided  by  1 2  is  equal  to  4. 

18^2  =  $4,  is  read  1 8  divided  by  2  is  equal  to  $4. 
9-^3  may  also  be  written  |,  where  9  is  the  dividend  and  3 
the  divisor. 

52.  When  the  divisor  does  not  exceed  12,  the  operation 
can  be  performed  mentally,  and  the  process  is  called  Short 
Division. 

When  all  the  different  steps  of  the  division  are  written, 
the  process  is  called  Long  Division. 

53.  Suggestions  to  the  Teacher.  —  Give  questions 
similar  to  the  following,  in  order  to  secure  facility  in  inter- 
preting results  and  accuracy  and  rapidity  in  using  the  multi- 
plication table. 

Thus  :  (1)  A  product  is  72  ;  one  factor  is  8.   Find  the  other. 
(2)  Whatisi36^4?     |72--i9?    172-^9?    132^11? 
Associate  simple  practical  questions  with  these  numbers. 


86  ARITHMETIC 

(3)  Extend  the  table  thus :  Divide  210  by  7  ;  3500  by  5  ; 
450  by  90  ;  840  by  12. 

(4)  Give  the  quotient  and  remainder  when  86  is  divided 
by  7  ;  93  by  12  ;  43  by  6. 

(5)  Reduce  to  the  next  higher  denomination  :  32  qt. ;  40  ^  ; 
96  in.;  45da. ;  450  min. 

(6)  The  unit  of  area  is  9  sq.  rd.  What  number  expresses 
the  ratio  of  the  area  of  a  field  containing  270  sq.  rd.  to  the 
unit  of  area  ? 

54.   If  1  T.  of  coal  costs  $  6,  how  many  tons  will  f  4764  buy  ? 

Here  the  product  is  4764  ;  one  factor  is  6,  and  we  are  required  to  find  the 
other  factor,  which  is  the  number  of  tons. 

6)4764 
794 

6  divides  47  of  the  hundreds'  unit  7  times  in  the  hundreds'  place,  with  a 
remainder  5  of  the  hundreds'  unit ;  5  of  the  hundreds'  unit  and  6  of  the  tens' 
unit  equal  56  of  the  tens'  unit. 

6  divides  56  of  the  tens'  unit  9  times  in  the  tens'  place,  with  a  remainder 
2  of  the  tens'  unit.  2  of  the  tens'  unit  and  4  of  the  one-unit,  equal  24,  which 
divided  by  6  equals  4. 

.*.  the  number  of  tons  =  794.- 

Exercise  63 

1.  Divide  each  of  the  following  numbers  by  3:  5187;  7864  j 
1783;  96,231;  52,867;  84,829. 

2.  Divide  by  4:  6552;  2496;  9897;  79,284;  70,837;  66,894. 

3.  Divide  by  5:  9565;  3127;  2704;  33,375;  80,346;  45,404. 

4.  Divide  by  6:  1698;  5934;  3353;  66,554;  46,893;  39,577. 

5.  Divide  by  7:  5964;  8828;  2495;  99,573;  87,049;  98,425. 

6.  Reduce  to  weeks :  273  da. ;  365  da. ;  4365  da. 

7.  Divide  by  8:  2564;  3683;  4992;  46,264;  84,364;  60,678. 


DIVISION  87 

8.  Divide  by  9:  3141;  6283;  6562;  15,708;  52,664;  82,315. 

9.  Divide  by  10:  3140;  6408;  2989;  43,825;  61,413;  84,375. 

10.  Divide  by  11:  1760;  5280;  4379;  30,005;  52,275;  65,341. 

11.  Divide  by  12:  2736;  5592;  1875;  28,060;  96,725;  10,008. 

12.  Divide  54,716  by  each  of  these  numbers :  3,  6,  9,  12. 

13.  Divide  63,360  by  each  of  these  numbers :  5,  7,  11,  12. 

14.  Divide  86,468  by  each  of  these  numbers :  4,  8,  10,  12. 

15.  Divide  75,918  by  each  of  these  numbers :  6,  8,  9,  11. 

16.  How  many  quarts  in  2  gal.  ?  A  city  milk  dealer  sold 
448  qt.  of  milk.  How  many  2  gal.  cans  are  required  to  hold  all 
the  milk  ? 

17.  A  dealer  sold  a  quantity  of  coal  for  $  6  a  ton,  and  received 
for  it  I  7950.     How  many  tons  did  he  sell  ? 

18.  A  merchant  sold  cloth  at  ^  3  a  yard,  and  received  for  it 
$  1344.     Find  the  number  of  yards. 

19.  A  rod  540  in.  long  has  a  piece  8  in.  long  cut  off  from  it, 
then  another  piece  of  the  same  length,  then  another,  and  so  on. 
How  often  may  this  be  done,  and  what  is  the  length  of  the  piece 
remaining  at  last  ? 

20.  $  15,108  was  paid  for  sheep  at  $  6  apiece.  Find  the  num- 
ber of  sheep. 

21.  What  number  must  be  added  to  91  to  make  it  exactly 
divisible  by  8  ? 

22.  The  expense  of  carpeting  a  room  was  $  45 ;  but  if  the 
breadth  had  been  3  ft.  less  than  it  was,  the  expense  would  have 
been  $  36.     Find  the  breadth  of  the  room. 

23.  The  expense  of  carpeting  a  room  was  $75;  but  if  the 
breadth  had  been  6  ft.  more  than  it  was,  the  expense  would  have 
been  $  105.     Find  the  breadth  of  the  room. 

24.  Find  the  number  of  strips  of  carpet  each  3  ft.  wide  re- 
quired to  carpet  a  room  15  ft.  wide;  21  ft.  wide;  27  ft.  wide; 
16  ft.  wide ;  6  yd.  wide ;  8  yd.  wide. 


88 


ARITHMETIC 


55.    (1)  Find  the  length  of  an  oblong  which  contains  96 
sq.  in.  and  is  8  in.  wide. 


Cut  off  from  the  oblong  a  strip  1  ft.  wide.    This  strip  contains  8  sq.  in. 

Here  we  are  given  the  whole  quantity,  or  96  sq.  in.,  and  the  measuring 
unit,  8  sq.  in.  The  number  12  gives  the  number  of  primary  units  of  1  in. 
contained  in  the  length.    Therefore  the  length  is  12  in. 

The  area  of  1  strip  1  in.  wide  =  8  sq.  in. 
The  number  of  strips  1  in.  wide  =  96  sq.  in.  -f-  8  sq.  in.  =  12. 
.-.  the  length  =  12  in. 

Hence  to  find  the  measure  of  the  length  divide  the  measure  of  the  area 
(96)  by  the  measure  of  the  width  (8).     Thus  96  ^  8  =  12. 

(2)  Find  the  number  of  yards  of  carpet  required  to  carpet 
a  room  32  ft.  long  and  26  ft.  wide,  the  carpet  running 
lengthwise,  if  each  strip  is  2  ft.  wide. 

The  number  of  strips  of  carpet  =  26  ft.  -r-  2  ft.  =  13. 
The  length  of  the  carpet  =  13  x  32  ft.  =  416  ft. 
=  138  yd.  2  ft. 
.  *.  138  yd.  2  ft.  of  cai-pet  are  needed. 

Make  a  diagram  for  this  question  on  the  scale  of  ^  in.  to 
1ft. 


DIVISION  89 

(3)  Reduce  6498  da.  to  weeks. 

In  this  question  the  time  is  expressed  in  terms  of  the  unit,  1  da.,  and  we 
are  required  to  express  it  in  terms  of  the  unit,  1  wk.  or  7  da.  Dividing 
6498  da.  by  7  da.,  the  result  is  928  wk.  2  da. 


Exercise  54 

In  the  following  exercise  prove  the  correctness  of  your  answers : 

1.  Reduce  to  quarts:  36  pt.  j  78  pt.  j  96  pt. ;  65  pt. ;  and 
257  pt. 

2.  Find  the  number  of  strips  of  carpet  2  ft.  wide,  required  to 
carpet  a  room  16  ft.  wide ;  20  ft.  wide ;  24  ft. ;  32  ft. 

3.  Reduce  to  yards:  384  ft.;  456  ft.;  723  ft.;  897  ft.;  5280  ft. 

4.  Find  the  number  of  strips  of  carpet  3  ft.  wide  required  to 
carpet  a  room  15  ft.  wide.  If  the  room  is  6  yd.  long,  how  many 
yards  are  needed  to  carpet  the  room? 

5.  Find  the  number  of  strips  of  carpet  2  ft.  wide  required  to 
carpet  a  room  16  ft.  wide.  If  the  room  is  8  yd.  long,  how  many 
yards  are  needed  to  carpet  the  room  ? 

6.  Make  a  drawing  (scale  1  in.  to  1  ft.)  to  show  how  many 
yards  of  carpet,  3  ft.  wide,  are  needed  to  carpet  a  room  12  ft.  wide 
and  15  ft.  long.     How  many  ? 

7.  Reduce  to  gallons  :  576  qt. ;  893  qt. ;  798  qt. ;  962  qt. 

8.  Divide  by  5  :    475 ;  827 ;  593 ;  890 ;  646. 

9.  Divide  by  6:    252;  435;  728;  846;  777. 

10.  Reduce  to  weeks:  245  da.;  365  da.;  678  da.;  899  da.; 
987  da. 

11.  Reduce  to  pecks:  32  qt.;  892  qt.;  958  qt.;  2456  qt. ; 
9472  qt. 

12.  Reduce  to  square  yards  :  756  sq.  ft. ;  894  sq.  ft. ;  3478  sq.  ft. ; 
9864  sq.  ft. 

13.  Reduce  to  dimes:  620^;  840^;  729^;  843^;  5246^;  8795^. 

14.  Divide  by  11:  451;  628;  847;  956;  8297;  7887. 


90 


ARITHMETIC 


15.  How  many  dozen  are  there  in  840  units  ?  957  units  ?  1459 
units  ?  4596  units  ? 

16.  Reduce  to  feet :  459  in.;  897  in.;  2641  in.;  63,360  in. 

17.  Find  the  number  of  square  inches  in  a  square  whose  side 
is  5  in.     Find  the  side  of  a  square  containing  64  sq.  in. 

18.  Find  the  lengths  of  the  sides  of  squares  whose  areas  are 
4,  25,  81,  36,  121,  49,  9,  100,  64,  and  144  sq.  in.  respectively. 

19.  Find  the  length  of  each  of  these  oblongs : 


Area 

Width 

Area 

Width 

12  sq.  in. 

Sin. 

45  sq.  ft. 

5  ft. 

24  sq.  in. 

4  in. 

240  sq.  yd. 

12  yd. 

72  sq.  in. 

Sin. 

144  sq.  mi. 

9  mi. 

20.   Find  the  length  of  a  room  12  ft.  wide,  which  contains 
192  sq.  ft. 

Exercise  55 
Find  the  quotient  and  remainder  and  prove  your  results  correct : 

1.  36-2;  48-3;  72-^4;  65^-5;  84 -- 6. 

2.  56-^7;  96-8;  81-9;  80-10;  77-11;  48^12. 

3.  249  H-  3 ;  842  -  6  ;  941  -  8 ;  654  -  8. 

4.  137-2;  439^5;  849-7;  999-12. 
11)682 

4)673 
10)6430 

2)2557 
12)8256 

8)7919 

5)9847 

8)7798 


5. 

6)743 

6. 

5)679 

7. 

7)2457 

8. 

6)3975 

9. 

6)4544 

10. 

11)8149 

11. 

4)2319 

12. 

11)8682 

9)847 

10)895 

8)976 

12)899 

9)1978 

12)4994 

11)7381 

3)4565 

5)1935 

4)3191 

9)4676 

7)6769 

6)1764 

12)9543 

9)7992 

12)63360 

DIVISION  91 

56.  (1)  A  father  dying  left  an  estate  valued  at  848,832  to 
be  divided  equally  among  his  wife,  his  two  sons,  and  his  four 
daughters.     What  was  the  share  of  each  ? 

In  this  problem  we  are  required  to  find  the  share  of  each,  which  is  the 
unit  of  measure.  In  order  to  find  this,  we  are  given  the  value  of  the  estate, 
which  is  the  whole  quantity,  and  one  factor,  which  is  the  number  of  shares  ; 
viz.  1  +  2  +  4,  or  7.  Therefore,  dividing  the  whole  quantity  by  7,  we  find 
the  share  of  each  to  be  |6976, 

(2)  The  area  of  the  four  walls  of  a  room  whose  dimensions 
are  8  yd.  and  6  yd.  is  112  sq.  yd.  Find  the  height  of  the 
room. 

We  are  here  required  to  find  the  height  of  the  room.  In  order  to  find  it 
we  are  given  the  area.  We  are  also  given  the  dimensions  with  which  we  can 
find  the  perimeter  of  the  room. 

Thus  we  may  think  of  the  area  of  the  walls  as  an  oblong  whose  sides  are 
the  perimeter  (28  yd.)  and  height,  and  area  112  sq.  yd.  Hence  the  measure 
of  the  height  =  112  -r-  28  =  4. 

.*.  the  height  =  4  yd. 


57.    (1)  245 


31  divides  75  of  the  hundreds'  unit  two  hundred  times.  Put  2  as  the  first 
term  in  the  quotient. 

Multiply  31  by  2,  and  subtract  the  product  62  from  75.  The  remainder  is 
13  of  the  hundreds'  unit.  Annex  the  9  tens  of  the  dividend,  making  139 
tens.  31  divides  139  tens  4  tens  times.  Put  4  as  the  second  term  in  the 
quotient.  Multiply  31  by  4,  and  subtract  the  product  124  from  139.  The 
remainder  is  15  of  the  tens'  unit.  Annex  the  8  units,  making  168  units.  31 
divides  l58  units  5  times.  Put  5  as  the  third  term  in  the  quotient.  Multiply 
81  by  5,  and  subtract  the  product  155  from  158.     The  remainder  is  3. 

What  is  the  quotient  on  dividing  7598  by  31  ?    What  does  it  show  ? 


92 


ARITHMETIC 


(2)   To   prove 
correct : 


that   the   answer  in  the  last   example  is 


.♦.  the  answer  is  correct. 


245  Quotient 
31  Divisor 

245 

735 

7595  Product 

3  Remainder 
7598  Dividend 

Or  thus,  by  division, 

31 

245)7598 
735 
248 

245 
3 

.  •.  245  quotient  and  3  remainder  is  the  correct  answer. 

58.   Divide  39,726  by  87. 
456 


87)39726 
348 
492 
435 


In  this  division  name  the  unit  to  which  each  remainder 
and  each  partial  dividend  belongs. 


576 
522 

54 


59.    Trial  divisor  and  trial  dividend. 

The  work  of  finding  the  quotients  can  be  much  simplified  by  using  the 
trial  divisor  and  trial  dividend. 

Thus  in  §  57,  as  31  is  nearer  30  than  40,  the  trial  divisor  is  3.  Dividing  3 
into  the  trial  dividends  7,  13,  and  15,  the  quotients  are  2,  4,  and  5. 

In  §  58,  as  87  is  nearer  90  than  80,  the  trial  divisor  is  9.  Dividing  9  into 
the  trial  dividends  39,  49,  and  57,  the  quotients  are  4,  5,  and  6. 

In  general,  if  the  divisor  is  61 .  62,  63,  615, 627,  or  634,  the  trial  divisor  is  6. 

If  the  divisor  is  67,  08,  69,  676,  689,  or  697,  the  trial  divisor  is  7. 


divisio:n-  93 

If  the  divisor  is  64,  65,  or  66,  the  use  of  the  trial  divisor  is  less  certain  ; 
but  the  rule  is  to  use  6  as  the  trial  divisor  for  64,  7  for  66,  and  either  6  or  7 
for  65. 

Name  the  trial  divisors  in  the  next  exercise. 

Exercise  56 

rind  the  quotients  and  remainders  of  the  following,  and  prove 
the  answers  to  the  odd  numbers  correct  by  multiplying,  and  the 
even  numbers  correct  by  dividing : 

1.  712 --31.  14.   7948 -V- 29.  27.   75,643^97. 

2.  2341-51.  15.  8543^49.  28.  23,877 -f- 24. 

3.  6287 -^  71.  16.  9765-5-69.  29.  38,753^34. 

4.  2195-70.  17.  8720-89.  30.  63,056-64. 

5.  5894  H- 91.  18.  8888^-38.  31.  74,111-25. 

6.  2068 -=-22.  19.  9894-18.  32.  96,433^75. 

7.  3572 -f- 42.  20.  9320-5-58.  33.  56,159^95. 

8.  1576 -^  62.  21.  16,324 -- 78.  34.  27,766 -r- 36. 

9.  8189 -H  82.  22.  30,086-98.  35.  56,139-^56. 

10.  6285^23.  23.  18,874 -j- 27.  36.  78,045^76. 

11.  7549-^53.  24.  21,803-5-37.  37.  204.80^32. 

12.  8476 -^  63.  25.  36,989^67.  38.  185.76 -^  24. 

13.  9989 -J- 93.  26.  52,298-87.  39.  27.144-36. 

Exercise  57 

1.  An  Illinois  farmer  raised  2784  bu.  of  corn  from  48  A. 
Find  the  number  of  bushels  per  acre. 

2.  A  farmer  raised  962  bu.  of  oats.  If  the  average  yield  per 
acre  was  37  bu.,  find  the  number  of  acres. 

3.  A  farmer  received  ^199.80  for  his  crop  of  wheat  at  74^  a 
bushel.  Find  the  number  of  bushels.  If  he  sowed  15  A.  with 
wheat,  find  the  average  yield  per  acre. 


94  ARITHMETIC 

4.  A  farmer  received  $158.08  for  his  crop  of  oats  at  38^  a 
bushel.  If  he  sowed  16  A.  with  oats,  find  the  average  yield  per 
acre. 

5.  How  many  pecks  in  2  bu.  ?  2  bu.  3  pk.  ?  3  bu.  1  pk.  ? 
6  bu.  2  pk.  ?  How  many  pecks  in  8  bu.  ?  48  bu.  ?  69  bu.  ? 
125  bu.?      336  bu.? 

6.  A  shippet  bought  225  bu.  of  apples  and  packed  them  in 
barrels  containing  2  bu.  2  pk.     Find  how  many  barrels  he  used. 

7.  A  man  bought  a  house  and  lot  for  $4500  and  paid  $3750 
in  cash.  If  he  paid  the  balance  in  monthly  instalments  of  $  25 
each,  in  how  many  months  would  he  pay  off  the  debt  ? 

8.  A  miller  put  up  3675  lb.  of  flour  in  49-lb.  sacks  and  sold 
them  at  $  1.10  each.     Find  the  total  selling  price. 

9.  A  grocer  bought  360  lb.  of  whitefish  in  15-lb.  pails  at  $1.55 
a  pail  and  sold  it  for  16  ^  a  pound.     Find  his  gain. 

10.  A  grocer  bought  864  lb.  of  tapioca  in  cases  containing  36 
one-pound  packages  at  $  3.30  a  case  and  sold  it  at  12  ^  a  pound. 
Find  his  gain. 

Exercise  58 

1.  10,377-^13.  9.  37,847 --86.  17.  854,300^49. 

2.  29,452-14.  10.  84,374 -f- 45.  18.  537,047 -- 36. 

3.  99,624  H- 15.  11.  22,158 --23.  19.  624,839^75. 

4.  87,643 --16.  12.  84,999 -- 69.  20.  802,666 --33. 

5.  63,277 -r- 17.  13.  15,273 -- 34.  21.  263,204-5-54. 

6.  64,935-18.  14.  42,965-88.  22.  467,989-6$. 

7.  99,658-19.  15.  335,296-^47.  23.  467,989^67. 

8.  29,943 -^  99.  16.  582,934-5-56.  24.  633,600^76. 

25.  604,826-29.  27.    494,358-65. 

26.  253,789 --96.  28.    832,016-79. 

In  the  following  exercise,  before  dividing,  make  a  careful  guess 
as  to  what  the  quotient  will  be. 


DIVISION  95 

Exercise  59 

1.  395,267 -- 105.  12.  $  367,989 -- 476. 

2.  300,498  --  207.  13.  578,243  cu.  iu.  --  231  cu.  in. 

3.  227,876  H- 121.  14.  578,243  rd.^  320  rd. 

4.  407,253  --  309.  15.  987,655  cu.  ft.  --  128  cu.  ft. 

5.  839,428-224.  16.  ^  128,821  ■- 360. 

6.  719,888-421.  17.  599,647-176. 

7.  584,287-593.  18.  313,947  da.  -  365  da. 

8.  495,638-784.  19.  444,555-366. 

9.  597,445  -  656.  20.  574,381  A.  -  640  A. 

10.  386,777  ^  921.  21.    987,432  sq.  in.  -  144  sq.  in. 

11.  811,394-675.  22.   358,049^528. 

Exercise  60 

1.  1  sq.  ft.  =?  sq.  in.  How  many  square  feet  in  864  sq.  in.  ? 
1728  ^q.  in.  ?     3456  sq.  in.  ?     5616  sq.  in.  ? 

2.  1  sq.  in.  =  640  A.  How  many  square  miles  in  a  section  of 
land  containing  1920  A.  ?    4480  A.  ?     16,00.0  A.  ?     23,680  A.  ? 

3.  A  section  of  land  containing  16,640  A.  is  drained.  Find  its 
area  in  square  miles. 

4.  1  gal.  contains  231  cu.  in.  Find  the  number  of  gallons  in 
1848  cu.  in. ;  3003  cu.  in. ;  8085  cu.  in. 

5.  How  many  gallons  will  a  tank  hold  that  contains  45,276 
cu.  in.  ? 

6.  1  cord  contains  128  cu.  in.  Find  the  number  of  cords  in 
1152  cu.  in. ;  9344  cu.  in. ;  35,328  cu.  in. 

7.  How  many  square  inches  in  an  oblong  4  in.  long  and  3 
wide  ?     8  in.  by  6  in.  ?     12  ft.  by  9  ft.  ? 

How  many  square  miles  in  a  section  of  land  12  mi.  long  and  6 
wide  ?  Into  how  many  parts  each  containing  8  sq.  mi.  can  it  be 
divided  ? 


96  ARITHMETIC 

8.  A  section  of  land  in  the  form  of  an  oblong  is  27  mi.  long 
and  12  mi.  wide.  Into  how  many  townships  each  containing 
36  sq.  mi.  can  it  be  divided  ? 

9.  The  freight  rates  from  Chicago  to  Hartford,  Conn.,  are  $  .55 
per  100  lb.  Find  the  number  of  sacks  of  flour,  each  weighing 
100  lb.,  that  can  be  sent  for  $  36.30.    Find  their  weight  in  pounds. 

10.  The  freight  rates  from  Chicago  to  Sioux  City,  Iowa,  are 
$  .48  per  100  lb.  Find  the  number  of  pounds  in  a  shipment  on 
which  the  freight  charges  were  $  35.52. 

Exercise  61 

1.  493,287  yd.  -  1760  yd.  8.  819,634-4972. 

2.  298,456  ft.  -  5280  ft.  9.  819,634-3264. 

3.  140,008  cu.  in.  - 1728  cu.  in.  10.  205,639-7459. 

4.  680,442 cu. in.- 2150 cu. in.  11.  726,998^9543. 

5.  998,209  lb.  -  2240  lb.  12.  337,877-^9961. 

6.  857,864  gr.  -  5760  gr.  13.  698,206-8456. 

7.  398,125  gr.  -  7000  gr.  14.  729,453^5879. 

Exercise  62 

1.  Find  the  wages  due  a  workman  who  has  worked  423  hr.  at 
$  1.50  a  day,  of  9  hr.  each. 

2.  A  man  sold  two  city  lots  for  $  1875  and  $  2125  respectively, 
and  used  the  money  to  buy  a  farm  of  80  A.  Find  the  cost  per 
acre. 

3.  The  temperatures  in  Chicago,  March  2,  1900,  are  given 
below : 

8  A.M 24      2  p.M 29 

10  A.M 26      4  p.M 30 

12  M 27      6  p.M 30 

8  p.M 30 

Find  the  average  temperature. 


DIVISION  97 

4.  March  2, 1900,  there  were  received  in  Chicago  27,200  bu.  of 
wheat.  At  340  bu.  to  the  car,  how  many  cars  of  wheat  were 
received  ?  At  40  cars  to  a  train,  how  many  train-loads  would  be 
necessary  to  bring  in  this  grain  ? 

5.  At  680  bu.  to  a  car  and  42  cars  to  a  train,  how  many  trains 
are  necessary  to  move  a  crop  of  6,854,400  bu.  ? 

6.  The  corn  crop  of  Kansas  for  1899  was  360,000,000  bu.  At 
400  bu.  to  a  car  and  40  cars  to  a  train,  how  many  trains  were 
necessary  to  move  this  crop  ? 

7.  The  area  of  Ehode  Island  is  1053  sq.  mi.,  and  in  1900  the 
population  was  428,556.  How  many  inhabitants  were  there  to 
the  square  mile  in  1900  ? 

8.  The  area  of  Massachusetts  is  8040  sq.  mi.,  and  in  1900  the 
population  was  2,805,346.  How  many  inhabitants  to  the  square 
mile? 

9.  Turn  to  your  geography  and  find  the  area  and  population 
of  several  states.  Divide  and  find  the  average  population  per 
square  mile.  Why  is  this  average  so  much  greater  in  some  states 
than  in  others  ? 

10.  For  the  35  weeks  ending  Feb.  26, 1900,  there  were  received 
in  the  city  of  St.  Louis  7,945,000  bu.  of  wheat.  Find  the  average 
number  of  bushels  per  week. 

11.  For  the  35  weeks  ending  Feb.  26,  1900,  there  were  received 
in  Toledo  10,654,000  bu.  of  wheat ;  in  Detroit,  2,359,000  bu. ;  in 
Kansas  City,  13,517,000  bu.  Find  the  average  number  of  bushels 
per  week  received  in  each  city. 

12.  How  many  years  from  the  beginning  of  the  year  1493  to 
the  close  of  the  year  1600  ? 

From  the  beginning  of  1493  to  the  close  of  1600,  $  501,640,000 
worth  of  gold  was  found  in  the  world.  Find  the  average  amount 
each  year. 

13.  State  how  to  divide  one  number  by  another  by  long  division. 
State  how  to  solve  each  of  the  preceding  questions  in  this 

exercise. 


98  ARITHMETIC 

60.   Divide  77.9G8  by  8. 

8)77.968  8  is  contained  in  77  units  9  times,  with  remainder  5  units. 

9.746     ^  ^®  contained   in   59  tentlis  7  tenths  times,  with  remainder 

3  tentlis.     8  is  contained  in  36  hundredths  4  hundredths  times, 

with  remainder  4  hundredths.    8  is  contained  in  48  thousaudtlis  6  thousandths 

times  with  no  remainder. 

Hence  the  operation  of  dividing  a  dividend  containing  a  decimal  is  similar 
to  that  of  dividing  when  the  dividend  does  not  contain  a  decimal.  Care 
must  be  taken  to  insert  the  decimal  point  in  the  quotient,  as  in  the  example, 
immediately  after  the  units'  figure  is  used  in  the  dividend. 

Exercise  63 
Divide : 

1.  12.6  by  6.  4.   239.76  by  37.        7.   195.2544  by  473. 

2.  7.56  by  7.  5.   596.36  by  17.        8.   192.4947  by  171. 

3.  16.38  by  13.       6.   889.92  by  72. 

9.  Divide  389.904  A.  of  land  equally  among  16  persons. 

10.  43  bu.  of  wheat  cost  ^37.625,  find  the  cost  of  1  bu. 

11.  25  bu.  of  corn  cost  $  8.625,  find  the  cost  of  1  bu. 

12.  94  bu.  of  oats  cost  $  22.09,  find  the  cost  of  1  bu. 

13.  Divide  the  following  numbers  by  10  : 

47.39        543.21         62        7.64 

14.  State  how  you  move  the  decimal  point  to  find  the  quotient 
on  dividing  a  number  by  10. 

15.  Write  down  the  quotients  obtained  on  dividing  the  follow- 
ing numbers  by  10 : 

64.52        3.9        742.63        95.614 

16.  State  how  to  find  the  quotient  without  actual  division  when 
a  decimal  is  divided  by  10. 

17.  Divide  by  100: 

792.6        8943.62        54.15        89467.1 

18.  State  how  to  find  the  quotient  without  actual  division  when 
a  decimal  is  divided  by  100.     By  1000. 


DIVISION  99 

19.  Write  down  the  quotients  obtcained  on  dividing  the  follow- 
ing numbers,  («)  by  10,  (b)  by  100 : 

24.3        2163.4        59.44        743.92        84        175        242 

20.  Write  down  the  quotients  obtained  on  dividing  the  follow- 
ing numbers,  (a)  by  1000,  (b)  by  100 : 

831.5        4926.4      .  2235         3852.1         4829        384 

Miscellaneous  Exercise  64 

1.  Divide  a  line  12  in.  long  into  parts  each  2  in.  long.     How 
many  parts  ?     12  in.  -h  2  in.  =  ? 

2.  Divide  a  line  1  ft.  long  into  parts  each  ^  ft.  long.     How 
many  parts  ?     1  f t.  --  i  ft.  =  ? 

3.  Make  drawings  to  show  the  value  of  1  ft.  -i-i  ft. ;  lit.-i-^  ft. ; 
1  ft.  -  1  ft. 

4.  Into  how  many  lots,  each  containing  J  A.,  can  you  divide 
1  A.? 

5.  A  society  divided  1  T.  of  coal  among  several  families,  giving 
to  each  J  T.     How  many  families  were  there  ? 

6.  Into  how  many  parts,  each  ^  ft.,  can  you  divide  a  line  1  ft. 
long  ?  3  ft.  long  ?  5  ft.  long  ? 

5  ft.  -  i  ft.  =  ?  4  gal.  -  J  gal.  =  ?  9  pk.  -  i  pk.  =  ? 

8  ft.  -  i  ft.  =  ?  6  bu.  -  i  bu.  =  ?  11  lb.  -r-  i  lb.  =  ? 

7  yd.  -  I  yd.  =  ?  12  da.  --  ^  da.  =  ?  ,12  mi.-^  mi.=  ? 

7.  A  lady  has  6  gal.  of  fruit  in  cans,  each  containing  -|-  gal. 
Find  the  number  of  cans. 

8.  Find  the  number  of  cans  in  each  case :  6  lb.  beef  in  ^  lb. 
cans  ;  8  lb.  extract  of  beef  in  ^  lb.  cans  ;  12  lb.  extract  in  i  lb.  cans. 

9.  Make  a  drawing  (scale  3  ft.  =  1  in.)  to  show  that  5  strips  of 
carpet  each  3  ft.  wide  will  be  needed  to  carpet  a  room  15  ft.  wide  ? 

10.  Find  the  number  of  strips  of  carpet  each  3  ft.  wide  required 
to  carpet  a  room  18  ft.  Avide ;  24  ft.  wide ;  26  ft.  wide ;  28  ft. 
wide.  If  the  room  is  in  each  case  10  yd.  long,  how  many  yards 
of  carpet  would  be  required  to  carpet  it  ? 


100  ARITHMETIC 

11.  How  many  yards  of  carpet  3  ft.  wide  are  needed  to  carpet 
a  room  12  ft.  wide  and  5  yd.  long  ? 

12.  Find  the  number  of  yards  of  carpet  3  ft.  wide,  needed  to 
carpet  each  of  the  following  rooms : 

Width  Length  *       Width  Length 

15  ft.  6  yd.  20  ft.  9  yd. 

18  ft.  8  yd.  18  ft.  24  ft. 

13.  Make  a  drawing  to  show  how  many  widths  of  matting 
J  yd.  wide  are  needed  to  cover  a  room  6  yd.  wide.  How  many  ? 
This  room  is  8  yd.  long,  how  many  yards  of  matting  are  needed  ? 

14.  Find  the  number  of  yards  of  matting,  ^  yd.  wide,  required 
to  cover  the  floors  of  each  of  the  following  rooms : 

Width  Length  Width  Length 

5  yd.  6  yd.  12  yd.  18  yd. 

6  yd.  8  yd.  16  yd.  18  yd. 
Find  the  cost  of  the  matting  for  the  last  room  at  24  ^  a  yard. 

15.  Find  the  cost  of  the  matting,  ^  yd.  wide,  needed  to  cover 
the  floors  of  a  hall  24  yd.  wide  and  30  yd.  long,  the  matting  cost- 
ing 28  ^  a  yard. 

16.  If  it  cost  $  49.60  to  carpet  a  room  with  carpet  48  ^  a  yard, 
what  wQuld  it  cost  with  carpet  at  60  ^  a  yard  ? 

17.  How  many  ounces  in  1  lb.  of  sugar  ?  Of  tea  ?  How  many 
pounds  in  32  oz.  ?  64  oz.?  96^oz.  ?  270  oz.  ?  736  oz.? 

18.  How  many  pounds  and  ounces  in  18  oz.  ?  24  oz.  ?  56  oz.  ? 
75  oz.  ?  300  oz.  ?  420  oz.  ?  648  oz.  ? 

19.  How  many  pounds  and  ounces  in  1  yd.  of  carpet  weighing 
17  oz.  to  the  yard  ?  19  oz.  ?  25  oz.  ?  35  oz.  ?  45  oz.  ? 

20.  Find  the  number  of  pounds  that  48  yd.  of  carpet  will  weigh 
at  29  oz.  per  yard. 

21.  I  bought  84  yd.  of  Wilton  carpet,  weighing. 45  oz.  per  yard. 
Find  its  weight. 

22.  The  following  live  stock  were  received  in  one  week  in 
Chicago : 


DIVISION 


101 


Cattle 

Calves 

Hogs 

Sii.:i:  > 

Wednesday,  Feb.  7  .     .     .     . 

17,261 

249 

47,019 

12,270 

Thursday,  Feb.  8       .     . 

8,775 

239 

36,089 

8,273 

Friday,  Feb.  9      .     .     . 

1,569 

212 

24,288 

4,529 

Saturday,  Feb.  10     .     . 

159 

6 

19,224 

1,505 

Monday,  Feb.  12  .     .     . 

19,831 

148 

42,705 

16,883 

Tuesday,  Feb.  13  .     .     . 

3,207 

807 

33,991 

13,275 

Wednesday,  Feb.  14 

16,000 

500 

36,000 

17,000 

Find  the  total  number  of  each  kind. 

23.    Find  from  the  following  table  the  increase  or  decrease  in 
the  price  in  each  of  the  following  articles  in  July,  1899 : 


Flour,  per  barrel  .... 
Lard,  per  pound  .... 
Beef  loins,  per  pound  .  .  . 
Beef  ribs,  per  pound  .  .  . 
Beef,  salt,  per  barrel  .  .  . 
Mess  pork,  per  barrel  .  .  . 
Bacon,  per  pound  .... 
Hams,  per  pound     .... 

Salt,  per  barrel 

Sugar,  granulated,  per  pound 

Thread,  spool ' 

Candles,  per  pound      .     .     . 
Coal,  anthracite,  per  ton  .     . 
Coal,  bituminous,  per  ton 
Matches,  parlor,  per  gross     . 

Nails,  per  keg 

Cement,  per  barrel .... 
Linseed  oil,  per  gallon      .     . 
Starch,  silver  gloss,  per  pound 
Leather,  per  pound      .     .     . 


Average  Whole- 
sale Price, 
January,  1890 


$5.50 

.056 

.19 

.16 

9.75 

10.00 
.056 

.lU 
.65 
.065 
.032 
.12 
4.20 
3.30 
4.75 
2.90 
.90 
.60 
.063 
.19 


Average  Whole- 
sale Price, 
July,  1899 


$4.50 

.049 

.21 

.17 
9.00 
8.75 

.052 

.10| 

.65 

.052- 

.031 

.12 
3.85 
2.20 
3.75 
2.70 

.80 

.39 

.058 

.22 


102 


ARITHMETIC 


24.  .  In  1898  the  cotton  crop  of  Missouri  was  1,247,128  bales, 
and  the  selling  price  $  25  a  bale.  In  the  next  year  it  was  1,212,- 
200  bales,  and  selling  price  $  11.25  greater  than  in  1898.  Find 
the  increase  in  the  value  of  the  crop. 

25.  The  fire  department  of  the  city  of  Chicago  asked  for  the 
following  sums  for  new  buildings  in  1902:  .f;  18,880;  $12,700; 
$  20,000 ;  $  22,100 ;  $  27,800 ;  $  23,400 ;  $  19,200.  Find  the  total 
amount. 

26.  The  following  table  shows  the  money  allowed  the  fire 
department  of  the  city  of  Chicago  for  1901  and  the  estimate  for 
1902: 


1901 


1902 


General  pay  roll     .... 
Fire  alarm  pay  roll     .     .     . 

Repairs 

Materials 

Repairs,  buildings,  and  boats 

Supplies 

Contingent  fund     .... 

Rent 

New  buildings 


1,427,664 

§1,445,084 

27,080 

24,230 

21),690 

40,000 

40,000 

25,000 

35,000 

148,000 

203,000 

1,200 

1,200 

11,856 

15,000 

18,382 

366,400 

Find  the  total  amount  in  each  case. 

27.  A  Chicago  daily  paper  states  that  there  is  a  net  increase  of 
$  466,000  in  the  estimate  of  the  fire  department  for  1902  over  1901. 
Show  that  there  is  an  error  of  $  4678  in  this  statement. 

28.  1  kilogram  =  220  lb.     1  bu.  of  wheat  weighs  60  lb. 

The  duty  on  wheat  imported  into  Germany  is  $  1.19  per  kilo- 
gram.    Find  the  duty  on  550  bu.  of  wheat. 


CHAPTER  YIII 

COMPARISON   OF   NUMBERS 

61.   If  $4  be  multiplied  in  turn  by  the  numbers  3  and  5, 

the  products  will  be  $  12  and  $  20.  Hence,  taking  1 4  as  the 
unit  of  measure,  and  noting  how  often  the  unit  is  repeated 
to  produce  $  12  and  $  20,  we  find  that  these  quantities  are 
represented  by  3  and  5  units  respectively.  Therefore  |  is 
the  ratio  of  $  12  to  f  20. 

What  is  the  ratio  of  $8  to  120?  Of  115  to  1 25?  Of 
$  25  to  $  15  ? 

Exercise  65 

1.  What  is  the  largest  unit  that  will  measure  $24  and  $30? 
How  often  in  each  case  ?     What  is  the  ratio  of  $  24  to  $  30  ? 

2.  What  is  the  largest  unit  that  will  measure  $20  and  $  35  ? 
What  is  the  ratio  of  $  20  to  $  35  ?     Of  $  35  to  $  20  ? 

3.  What  is  the  ratio  of  $  24  to  $  36  ?  $  36  to  $  24  ?  $  27 
to  $30?     $30  to  $27?     $55  to  $60?     $60  to  $36? 

4.  What  is  the  ratio  of  14  bu.  to  35  bu.  ?  49  bu.  to  63  bu.  ? 
32  lb.  to  56  lb.  ?     54  min.  to  36  min.  ?     55  T.  to  66  T.  ? 

5.  What  is  the  ratio  of  the  value  of  48  bbl.  of  flour  to  60  bbl. 
of  flour  ?  Of  24  bu.  of  oats  to  42  bu.  of  oats  ?  Of  70  bu.  of 
wheat  to  49  bu.  of  wheat  ? 

6.  What  is  the  ratio  of  36  lb.  to  45  lb.  ?  45  lb.  of  tea  cost 
$  30 ;  what  part  of  $  30  will  36  lb.  cost  ?     How  much  ? 

7.  What  is  the  ratio  of  14  to  21  ?  If  21  T.  of  hay  cost  $180, 
what  will  14  T.  cost? 

103 


104  ARITHMETIC 

8.  If  36  yd.  of  cloth  cost  $  28,  what  will  27  yd.  of  the  same 
kind  cost  ? 

9.  If  27  T.  of  coal  cost  $  180,  what  will  36  T.  cost  ? 

10.  What  is  the  ratio  of  9  to  12  ?     Of  12  to  9  ? 

9  men  can  do  a  piece  of  work  in  24  da.  Should  you  multiply 
24  da.  by  |  or  by  f  to  find  how  long  it  would  take  12  men  to  do 
the  same  piece  of  work  ?     How  many  days  ? 

11.  12  men  can  do  a  piece  of  work  in  18  da.  Should  you 
multiply  18  da.  by  f  or  |  to  find  how  long  it  would  take  9  men 
to  do  the  same  piece  of  work  ?     How  many  days  ? 

12.  If  15  men  can  dig  a  ditch  in  12  da.,  how  long  will  it  take 
20  men  to  dig  a  ditch  of  the  same  size  ? 

62.   (1)  What  is  the  ratio  of  1 20  to  $  45  ? 

Let  $  5  be  taken  as  the  unit  of  measure. 
Then  $  20  is  measured  by  4  times  the  unit. 
And  $  45  is  measured  by  9  times  the  unit. 
.-.  $  20  is  I  of  $  45. 

(2)    If  22  yd.  of  cloth  cost  116,  what  will  33  yd.  cost  at 

the  same  rate  ? 

Take  11  yd.  as  the  unit  of  length. 

Then  33  yd.  =  |  of  22  yd. 

.-.  33  yd.  cost  |  of  $  16  or  $24. 

Exercise  QQ 

1.  What  is  the  ratio  of  ^  18  to  $  24  ?     $  35  to  ^  55  ?     f  28 
to  ^  63  ? 

2.  What  is  the  ratio  of  16  hr.  to  56  hr.  ?     72  hr.  to  45  hr.  ? 

3.  What  is  the  ratio  of  60  mi.  to  25  mi.  ?     99  A.  to  55  A.  ? 

4.  If  45  cd.  of  wood  cost  $  162,  what  will  20  cd.  cost  ? 

5.  If  21  T.  of  hay  cost  $  174,  what  will  70  T.  cost  ? 

6.  If  36  yd.  of  cloth  cost  $  42,  what  will  24  yd.  cost  at  the 
same  rate  ? 


COMPARISON   OF   NUMBERS  105 

7.  A  miller  sold  35  bbl.  of  flour  for  $  126.     How  much  will 
he  receive  for  15  bbl.  at  the  same  rate  ? 

8.  A  train  runs  32  mi.  in  48  min.     At  the  same  rate,  what 
distance  will  it  run  in  54  min.  ? 

9.  If  84  men  can  dig  a  trench  in  36  da.,  how  long  will  it  take 
108  men  to  dig  a  trench  of  the  same  size  ? 

10.  If  88  horses  eat  33  bu.  of  oats  in  1  da.,  how  many  bushels 
will  48  horses  eat  in  the  same  time  ? 

11.  If  45  men  can  reap  a  field  of  36  A.  in  a  certain  time,  how 
many  acres  would  30  men  reap  in  the  same  time  ? 

12.  A  bankrupt  pays  $35  out  of  every  $  63  owed.  How  much 
shall  I  receive  if  he  owes  me  $  81  ? 

13.  If  32  T.  of  coal  cost  $  184,  what  will  88  T.  cost  at  the 
same  rate  ? 

14.  If  56  men  can  do  a  piece  of  work  in  21  da.,  how  long  will 
it  take  24  men  to  do  it  ? 

15.  How  many  pounds  of  tea  can  be  bought  for  $56,  at  the 
rate  of  $  16  for  34  lb.  ? 

16.  Tea  is  bought  at  72^  a  pound,  and  sold  for  84^  a  pound. 
The  gain  is  what  part  of  the  cost  price  ? 

17.  The  cost  of  fencing  132  rd.  of  railway  is  $  117.  What  is 
the  cost  of  fencing  88  rd.  ? 

18.  If  a  12-qt.  pail  is  just  filled  by  6  units  of  milk,  how  many 
quarts  are  there  in  a  pail  which  will  hold  5  units  ?  What  is  the 
unit  ? 

19.  Fifty-four  minutes  are  represented  by  9  units  of  time. 
How  many  minutes  are  there  in  7  units  ? 

20.  If  63  men  can  dig  a  trench  in  16  da.,  how  long  will  it  take 
18  men  to  dig  it  ? 


CHAPTER   IX* 

SQUARE  ROOT 

63.  Show  by  drawing  a  square  whose  side  is  3  in.  that  a 
3-in.  square  contains  9  sq.  in.  How  many  square  inches 
does  a  4-in.  square  contain  ?     A  5-in.  square  ? 

64.  The  product  of  3  and  3  is  9  ;  of  5  and  5  is  25.  The 
squares  whose  sides  measure  3  and  5  units  of  length  contain 
9  and  25  units  of  square  measure.  We  say  that  9  is  the 
square  of  3  and  that  25  is  the  square  of  5;  that  3  is  the 
square  root  of  9  and  5  the  square  root  of  25. 

The  square  of  3  is  written  3^,  and  the  square  root  of  9  is 
indicated  thus :  V9. 

2  is  called  the  Exponent,  and  ^  the  Radical  Sign.  3^  is  also 
called  the  second  Power  of  3. 

65.  The  Square  of  a  number  is  the  product  found  by  mul- 
tiplying the  number  by  itself. 

Thus  the  squares  of  1,  2,  3,  4,  5,  6,  7,  8,  9,  10, 
are  1,  4,  9,  16,  25,  36,  49,  64,  81,  100. 

66.  The  square  root  of  a  number  is  one  of  its  two  equal 
factors. 

Thus  the  square  roots  of  1,  4,  9,  16,  25,  36,  49,  64,  81, 100, 
are  1,  2,  3,    4,    5,    6,    7,    8,    9,    10. 

*  Note.  —  If  proferred,  this  chapter  may  be  postponed  until  Chapter  WIT 
is  reached. 

106 


SQUARE  ROOT  107 

How  many  digits  in  the  square  root  of  a  number  contain- 
ing one  or  two  digits  ? 

67.     Pupils  should  memorize  the  tables  in  the  two  preceding  paragraphs 
and  be  able  to  answer  instantly''  such  questions  as  the  following : 

What  is  the  first  figure  in  the  square  root  ol  27  ?     68  ?      76?     43  ?      80  ? 

Exercise  67 

Write  the  following  products  as  powers: 
1.    5  X  5.  2.    7  X  7.  3.    24  x  24. 

Write  the  following  powers  as  products  and  find  their  values : 

4.  81  6.    461  8.    (if. 

5.  131  7.    (1)^.  9.    (11)1 

Prove  the  following  statements  by  multiplication : 

10.    V49  =  7(7x7=:?).    14.    V324  =  18.         18.    V|||  =  if 


11.  V169  =  13.  15.    V961  =  31.        19.    Villi  =  f J. 

12.  V729  =  27.  16.    V2025  =  45.       20.    V6l  =  2i. 


13.    V6.25  =  2.5.  17.    V23.04  =  4.8.     21.    V53i  =  7f 

Exercise  68 

1.  The  square  of  a  number  of  one  digit  contains  how  many 
digits  ?     (See  §  65.) 

2.  Find  the  squares  of  the  numbers  from  10  to  20  and  commit 
the  results  to  memory. 

3.  Find  the  squares  of  25,  28,  54,  75,  and  99. 

4.  From  the  results  obtained  in  examples  2  and  3,  state  how 
many  digits  are  found  in  the  square  of  a  number  of  2  digits. 

5.  Find  the  squares  of  175,  199,  246,  402,  814,  999. 

6.  From  the  results  in  Example  5  state  how  many  digits  are 
found  in  the  square  of  a  number  of  3  digits. 


108  ARITHMETIC 

7.  Judging  from  the  results  obtained  in  Examples  2,  3,  and  5, 
state  the  number  of  digits  in  the  square  root  of  a  square  number 
that  contains  3  digits ;  4  digits ;  6  digits  ;  7  digits ;  8  digits. 

8.  How  many  digits  in  the  square  correspond  to  1  digit  in 
the  square  root  ? 

9.  What  is  the  square  root  of  400  ?     Of  900  ? 

10.  The  square  root  of  625  lies  between  what  two  numbers  ? 

11.  Find  the  square  of  10,  20,  30,  40,  50,  60,  70,  80,  and  90. 

12.  Between  what  numbers  does  the  square  root  of  1225  lie? 
Of  4225?      Of  2304?      Of  8281  ?      Of  8704? 

13.  What  is  the  square  of  200  ?      Of  300  ? 

14.  The  square  root  of  71,289  lies  between  what  two  numbers  ? 

15.  Find  the  squares  of  100,  200,  300,  etc.,  up  to  900. 

16.  Between  what  two  numbers  does  the  square  root  of  271,441 
lie  ?      Of  795,664  ? 

68.   The  following  explanation  will  make  clear  the  metliod 
of  finding  the  square  root  of  a  number  of  3  or  4  digits. 

24  Thus  24,  which  is  made  up  of  two  parts,  20  and  4,  has  for  its 

QA  square  576,  which  is  seen  to  be  made  up  of  400,  the  square  of  20  ; 

—  16,  the  square  of  4 ;  and  twice  the  product  of  20  and  4. 

^^  Now  to  recover  24  from  576,  we  know  that  its  hundreds'  digit  6, 

80  showing  that  the  number  is  between  400  and  000,  gives  the  tens'  digit 

80  of  the  root,  so  that  we  know  one  of  the  parts  of  the  root,  viz.,  20.    The 

AQf\  square  of  20  is  400,  and  the  rest  of  the  given  number,  176,  must  be 

— —  2  times  20,  multiplied  by  the  other  part,  together  with  the  square  of 

'^'^  the  other  part.     Multiplying  20  by  2,  and  using  the  product  40  as  a 


20 

40 


^7rr20  -I-  4       divisor  with  176  as  dividend,  we  get  the  quotient  4. 
'     ^  Multiplying  40  by  4  and  subtracting  the  product  100 

'*^^  from  176,  we  get  the  remainder  16, 


176  which  is  the  square  of  4.    Therefore         2 

1  f)0  20  +  4  or  24  is  the  square  root  of  576. 

-j /.  The  work  of  extracting  the  square      ^^ 

root  may  be  simplified  by  leaving  out 

•^^  the  unnecessary  zeros,  thus  ; 


5'76(24 
4 


176 
176 


SQUARE   ROOT 


109 


69.    (1)  The  method  of  discovering  the  square  root  of  a 
number  of  5  or  6  digits  is  similar  to  that  for  finding  the 
square  root  of  numbers  of  3  or  4  digits. 
246 
246 
36 
1440 
1440 
57600 


Thus  246,  which  is  made  up  of  two  parts,  240  and  6,  has  for 
its  square  60,516,  which  is  seen  to  be  made  up  of  57,600,  the  square 
of  240  ;  36,  the  square  of  6  ;  and  twice  the  product  of  240  and  6. 


60516 

2 

6^051 

4 

44 

205 

176 

480 

2916 

2880 

6 

36 

36 

44 


486 


6'05'16(246 
4 


205 
176 


1-0  _|_  g  (2)    Hence  proceeding  with  605  as  in  §  68 

with  576,  we  get  in  tlie  square  root  24  tens  or 
240.     Multiplying  240  by  2  and  using  the  prod- 
uct 480  as  a  divisor  with  2916  as  a  dividend,  the 
quotient  is  found  to  be  6. 
Multiplying  the  480 
by  6,   and   subtracting 
the  product  2880  from 
2916   we  have   the   re- 
mainder   36,   which    is 
the  square  of  6.    There- 
fore 240  +  6  or  246  is  the  square  root  of  60516. 

Leaving  out  the  unnecessary  zeros,  the  work  may 
be  simplified  as  in  the  contracted  form. 

The  number  whose  square  root  is  to  be  extracted  should  be  pointed  off 
into  groups  of  two  figures,  as  in  the  preceding  examples,  beginning  with  the 
units'  figure. 

70.   To  find  the  square  root  of  ||f . 

These  roots  of  289  and  625  are  found  to  be  17  and  25. 

Hence  Vffl  =  H- 

Exercise  69 


2916 
2916 


Find  the  square  root  and  prove  your  answer  correct : 

1.  324.  5.    3025.  9.    71,824.  13. 

2.  529.  6.    6889.  10.    101,761.  14. 

3.  841.  7.    4096.  11.   465,124.  15. 

4.  1156.  8.    9409.  12.    998,001.  16. 


64 

■289"- 
_729 
"2T0"T- 

301 


22 


16* 


110  ARITHMETIC 

17.  Compare  the  process  of  extracting  the  square  root  of  a 
number  with  that  of  long  division. 

Find  the  square  root  of : 

18.  3136.  20.    .4096.  22.    219.04.  24.    1176.49. 

19.  72.25.  21.    7.8961.  23.    29.9209.  25.    .469225. 

71.   (1)  Find  the  side  of  a  square  containing  4225  sq.  in. 

"The  square  is  measured  by  4225  units  of  1  sq.  in. ;  therefore  the  side  is 
measured  by  V'4225,  or  65  units  of  1  in. ,  i.e.  the  length  of  the  square  is  65  in. 

(2)  The  sides  of  a  rectanguhir  field  containing  735  sq.  rd. 
are  as  3  to  5.     Find  their  length. 

The  field  contains  3  x  5,  or  15  units  of  area. 
The  area  of  one  unit  =  735  sq.  rd.  -4-  15,  or  49  sq.  rd. 
The  side  of  a  square  containing  49  sq.  rd.  =  7  rd. 
.-.  the  sides  of  the  field  are  3  x  7  rd.,  or  21  rd.,  and  5  x  7  rd.,  or  36  rd. 
Draw  a  rectangle  whose  sides  are  as  3  to  5,  and  divide  it  into  15  squares  to 
illustrate  this  example. 

Statement  of  Solution 

First  find  the  number  of  units  of  area  in  the  field.  (How  many  ?)  Divide 
this  number  into  the  area,  and  find  the  unit  of  area.  (What  is  it?)  Then 
find  the  unit  of  length,  which  is  the  length  of  the  side  of  the  unit  of  area. 
(What  is  it  ?) 

Multiply  the  unit  of  length  by  3  and  5  respectively,  to  find  the  sides  of  the 
field. 

(3)  To  find  the  area  of  a  triangle^  the  lengths  of  whose  sides  are  given, 
find  one-half  the  sum  of  the  number  of  units  of  length  in  the  sides.  Stibtract 
from  this  the  number  of  units  of  length  in  each  side  separately.  Find  the 
product  of  these  four  results.  The  square  root  of  this  product  is  the  mimber 
of  units  of  area  in  the  given  triangle. 

Find  the  area  of  a  triangle  whose  sides,  are  5  in.,  12  in., 
and  13  in.  respectively. 

The  sum  =  5  +  12  +  13  =  30. 
One-half  this  sum  =  15. 

15  -    5  =  10.  15  X  10  X  3  X  2^  900. 
15-12=    3.  \/900  =  .30. 

16  —  13  =   2.  /.  Uie  area  of  the  triangle  =  30  sq.  in. 


SQUARE  KOOT  111 

Exercise  70 

1.  Find  the  length  of  the  side  of  a  square  park  containing 
5625  sq.  yd. 

2.  How  many  rods  are  there  in  the  side  of  a  square  field  con- 
taining 1156  sq.  rd.  ? 

3.  How  many  square  inches  in  an  oblong  9  in.  wide  and 
16  in.  long  ?     How  many  inches  in  a  square  of  the  same  size  ? 

4.  A  flower  bed  is  9  ft.  long  and  4  ft.  wide.  Find  the  side  of 
a  square  flower  bed  of  the  same  size. 

5.  Find  the  side  of  a  square  that  shall  contain  as  many  square 
feet  as  an  oblong  264  ft.  long  and  66  ft.  wide. 

6.  7  sq.  yd.  1  sq.  ft.  =  ?  sq.  ft. 

Find  the  side  of  a  square  containing  7  sq.  yd.  1  sq.  ft. 
Find  the  side  of  a  square  containing  13  sq.  yd.  4  sq.  ft. 

7.  Find  the  length  of  the  side  of  an  enclosure  in  the  form  of 
a  square  containing  386  sq.  yd.  7  sq.  ft. 

8.  A  garden  bed  contains  48  sq.  ft,,  and  it  is  3  times  as  long 
as  it  is  wide.  Make  a  drawing  of  the  bed.  Find  its  length  and 
width. 

9.  A  park  contains  9408  sq.  yd.,  and  it  is  3  times  as  long  as  it 
is  wide.     Find  its  length  and  width. 

10.  A  boy  bought  a  number  of  oranges,  paying  as  many  cents 
for  each  orange  as  there  were  oranges.  All  cost  25  ^.  How  many 
oranges  did  he  buy,  and  at  what  price  for  one  orange  ? 

11.  A  merchant  bought  a  number  of  yards  of  cloth,  paying  as 
many  cents  for  each  yard  as  there  were  yards.  The  entire  cost 
was  $  56.25.  How  many  yards  did  he  buy,  and  at  what  price 
per  yard? 

12.  What  is  one  of  the  two  equal  factors  of  36  ?  Of  64  ?  Of 
121? 

13.  What  is  one  of  the  two  equal  factors  of  24,336  ? 

14.  A  rectangular  field,  the  sides  of  which  are  in  the  ratio  of 
4  to  7,  contains  4032  sq.  rd.     Find  the  length  of  each  side. 


112  ARITHMETIC 

15.  Find  the  perimeter  of  the  field  in  the  previous  example 
and  the  cost  of  fencing  the  field  at  $  4  a  rod. 

16.  A  body  of  soldiers  in  column  form  567  ranks,  7  abreast. 
If  they  were  drawn  up  in  solid  square,  how  many  would  there  be 
on  each  side  ? 

17.  Find  the  side  of  a  square  which  is  equal  in  area  to  the  sum 
of  the  area  of  two  squares,  the  sides  of  which  are  6  in.  and  8  in. 
long. 

18.  Draw  two  lines  respectively  6  and  8  in.  long,  at  right 
angles.  Join  their  extremities  by  a  straight  line.  Measure  this 
line  and  show  that  it  is  equal  to  the  side  of  the  square  found  in 
question  17. 

•  19.  Work  examples  similar  to  17  and  18,  using  the  following  as 
the  lengths  of  the  sides  of  the  smaller  squares :  3  in.,  4  in. ;  5  in., 
12  in. ;  8  in.,  15  in. 

20.  From  the  preceding  three  questions  make  a  rule  showing 
how  to  find  the  length  of  the  hypotenuse  of  a  right  triangle  when 
the  lengths  of  the  other  tWo  sides  are  known. 

21.  What  is  the  hypotenuse  of  a  right  triangle  whose  sides  are 
21  ft.  and  28  ft.  ?     15  ft.,  36  ft.  ?     56  ft.,  105  ft.  ? 

22.  Find  the  side  of  a  square  equal  in  area  to  the  difference  of 
the  area  of  the  two  squares  whose  sides  are  41  ft.  and  9  ft. 

23.  What  is  the  altitude  of  a  right  triangle  whose  hypotenuse 
and  base  are  34  ft.,  16  ft.  ?     205  ft.,  45  ft.  ?     136  ft.,  64  ft.  ? 

24.  The  top  of  a  ladder  rests  against  the  side  of  a  building 
84  ft.  from  the  ground,  and  its  foot  is  35  ft.  from  the  wall.  Find 
the  length  of  the  ladder. 

25.  A  ladder  51  ft.  long  stands  close  against  a  building.  How 
far  must  the  foot  be  drawn  out  that  the  top  may  be  lowered  6  ft.  ? 

26.  Find  the  diagonal  of  a  rectangular  field  whose  sides  are 
144  yd.  and  60  yd. 

27.  Find  the  side  of  a  square  equal  in  area  to  a  rectangle  whose 
sides  are  148  yd.  and  333  yd.  Find  the  difference  between  the 
perimeters  of  the  rectangle  and  square. 


SQUARE   ROOT  113 

28.  Find  the  area  of  the  largest  rectangle  which  can  be  enclosed 
by  a  line  36  in.  long. 

29.  A  field  in  the  form  of  a  rectangle  whose  sides  are  as  3  to  4 
contains  432  sq.  rd.  How  much  do  I  save  by  crossing  along  its 
diagonal  instead  of  going  along  its  two  sides  ? 

30.  The  sides  of  a  triangle  are  6  in.,  8  in.,  and  10  in.  Find  its 
area.  Draw  the  triangle.  Does  your  result  seem  to  correspond 
to  the  drawing  ? 

31.  The  sides  of  a  triangle  are  8  in.,  15  in.,  and  17  in.  Find  its 
area. 

32.  Find  the  area  of  a  triangle  whose  sides  are  12  in.,  16  in., 
and  20  in. 

3.    Find  the  areas  of  the  triangles  whose  sides  are: 
21  in.,  28  in.,  35  in.  4  in.,  7.5  in.,    8.5  in. 

24  in.,  45  in.,  51  in.  3  in.,  2.4  in.,    1.8  in. 

9  in.,  40  in.,  41  in.  4  in.,  9.6  in.,  10.4  in. 

Miscellaneous  Exercise  71 

1.  3  ft.  H-Jft.  =  ?  4  1b. -^Ib.  =  ?  6T. -^T.  =  ? 

2.  Draw  a  line  4  ft.  long,  and  show  that  it  can  be  divided  into 
6  parts,  each  |  ft.  long.     4  f t.  ^  f  ft.  =  ? 

3.  Make  drawings  to  show  that : 

(1)  6  ft.  -  I  ft.  =  9.  (2)  6  ft.  -  I  ft.  =  8. 

4.  Into  how  many  parts,  each  }  ft.  long,  can  you  divide  a  line 
6  ft.  long  ?  How  many  of  these  24  parts  make  f  ft.  ?  How  many 
parts,  each  equal  to  |  ft.,  can  you  count  in  these  24  parts  ? 

5.  In  the  second  part  of  example  3,  by  what  do  you  multiply 
6  to  get  24  ?  By  what  do  you  divide  to  get  8  ?  How,  then,  do 
you  obtain  the  result  on  dividing  6  by  J  ?  Is  it  correct  to  divide 
by  3  first,  and  then  multiply  by  4  ?  Which  gives  the  smaller 
numbers  to  work  with  ? 


114  ARITHMETIC 

6.  How  do  you  divide  6  by  |  ?     8  by  f  ?    How  do  you  divide 
any  whole  number  by  a  fraction  ?     8  h-  |  =  8  x  J  =  10. 

7.  Find  the  quotient  in  each  of  the  following : 


4  ft. 

^|ft. 

36  A.  -  f  A. 

63 -H 

9  yd. 

^fyd. 

32  A.  -  f  A. 

51-^4i 

12  1b. 

-fib. 

42yr.-2iyr. 

35 -If 

10  1b. 

-fib. 

72  yr.  ~  2\  yr. 

98 -8J 

14  T. 

^-|T- 

84yr-2fyr. 

62H-4f 

8.  How  many  yards  of  cloth  will  cost  $  6  at  $  J  a  yard  ?  At 
f  f  a  yard  ?     At  $li  a  yard  ? 

9.  Make  a  drawing  to  show  how  many  strips  of  carpet  |  yd. 
wide  are  needed  to  carpet  a  room  6  yd.  wide  and  8  yd.  long. 
(Scale  1  in.  to  1  yd.)     How  many  strips  ?     How  many  yards  ? 

10.  How  many  strips  of  carpet  |  yd.  wide  are  needed  to  car- 
pet a  room  12  yd.  wide  ?  If  this  room  is  16  yd.  long,  how  many 
yards  of  carpet  are  needed  to  carpet  the  room  ? 

11.  How  many  yards  of  carpet  f  yd.  wide  are  needed  to  carpet 
a  room  9  yd.  wide  and  12  yd.  long  ?     Find  its  cost  at  75^  a  yard. 

12.  How  many  square  yards  in  a  piece  of  floor  oil  cloth,  weigh- 
ing 14  lb.,  at  3^  lb.  a  square  yard  ? 

13.  A  wool  carpet  weighs  1|  lb.  to  the  yard.  How  many  yards 
in  a  roll  weighing  18  lb.  ? 

An  ingrain  carpet  weighs  1 J  lb.  to  the  yard.  How  many  yards 
in  a  roll  weighing  24  lb.  ? 

14.  How  many  silver  spoons  weighing  IJ  lb.  per  dozen  will 
weigh  10  lb.  ? 

How  many  silver  spoons  weighing  IJ  lb.  per  dozen  will  weigh 
12  lb.  ? 

15.  A  double  roll  of  wall  paper  weighs  1|  lb.  How  many 
double  rolls  weigh  21  lb.  ?     Find  its  value  at  $  .45  a  double  roll. 


SQUARE  ROOT 


115 


16.    The  strength  of  the  American  and  British  navies  is  given 
below.     Find  the  total  number  of  ships  in  each  case. 


Battleships     .     .     . 
Armored  cruisers    . 
Protected  cruisers  . 
Unprotected  cruisers 
Coast-defence  ships 
Torpedo  vessels .     , 
Ships  for  special  purposes 
Torpedo-boat  destroyers 
Torpedo  boats    .     .     . 


Completed 


U.S. 


5 

2 
14 
10 
20 

1 

2 


Eng. 


52 

18 
95 
16 
15 
35 
3 
50 


Under  Construction 


U.  S. 


20 

22 


Eng. 


12 

8 

24 


46 


17.   Find  the  increase  or  decrease  in  the  relative  price  of  the 
following  on  July  1,  1899 : 


Relative 
Price 
Jan.  1, 

1890 

Rela- 
tive 
Price 
July  1, 
1899 

Relative 

Price 

Jan.  1, 

1890 

Rela- 
tive 
Price 

July  1, 

1899 

Beans         .     .     . 

86.9 

64.5 

Beef 

97.5 

105.7 

Bread    .... 

97.1 

97.8 

Salt  meat 

93.8 

82.8 

Butter  .... 

106.0 

87.5 

Bacon    . 

89.5 

82.5 

Cheese  .... 

107.1 

86.9 

Ham 

100.5 

101.8 

Coffee    .... 

101.8 

36.2 

Mutton . 

104.7 

78.5 

Eggs      .... 

107.4 

70.9 

Milk.     . 

105.5 

88.0 

Fish 

99.9 

90.7 

Molasses 

114.6 

99.6 

Flour     .... 

92.7 

75.8 

nice  .     . 

94.0 

105.7 

Fruit  (preserved) 

93.8 

73.1 

Salt  .     . 

94.3 

90.4 

Lard      .... 

92.6 

80.5 

Sugar     . 

117.0 

102.6 

Cornmeal  .     .     . 

82.0 

66.5 

116  ARITHMETIC 

18.  Find  the  cost  of  562  T.  of  hay  at  $  11.50  a  ton. 

19.  How  many  pounds  does  1  bu.  of  wheat  weigh  ?  What  is 
the  ratio  of  its  weight  to  100  lb.  ?  If  the  freight  rate  on  wheat 
is  10  ^  per  100  lb.,  what  is  the  rate  per  bushel  ? 

20.  If  the  freight  rate  on  wheat  is  15^  per  100  lb.,  what  is  the 
cost  of  shipping  3000  bu.  of  wheat  ? 

21.  Find  quotients  and  remainders: 

82963  --  48  60935  --  66  889966  -r-  96 

70400  --  24  10004  -v-  82  384729  h-  99 

22.  Find  the  square  root  of: 

576  103041  692224 

1849  10.3041  6922.24 

23.  Find  the  square  root  of:  ^f;  |||;  V;  ^1;  '^i;  ^^' 

24.  Find  the  side  of  a  square  lot  that  shall  contain  as  many 
square  feet  as  a  lot  150  ft.  long  and  96  ft.  wide. 


CHAPTER   ?: 

GREATEST    COMMON    MEASURE   AND    LEAST    COMMON 

MULTIPLE 

72.  Name  all  the  units  of  length  which  will  exactly  meas- 
ure 15  in. 

They  are  1  in.,  3  in.,  5  in.,  and  15  in. 

73.  Find  all  the  different  units  of  length  that  will  exactly 
measure  12  ft.  and  18  ft. 

The  measures  of  12  ft.  are  1,  2,  3,  4,  6,  and  12  ft. 

The  measures  of  18  ft.  are  1,  2,  3,  6,  9,  and  18  ft. 

It  is  evident  that  all  the  common  measures  of  12  ft.  and 
18  ft.  are  1,  2,  3,  and  6  ft.,  and  that  the  greatest  commqn 
measure  is  6  ft. 

A  Common  Measure  of  two  or  more  quantities  is  a  unit  that 
will  exactly  divide  each  of  them. 

The  Greatest  Common  Measure  (G.  CM.)  of  two  or  more 
quantities  is  the  largest  unit  which  will  exactly  divide  each 
of  them. 

For  convenience  we  speak  of  the  common  measure  or  the 
greatest  common  measure  of  two  or  more  numbers. 

74.  Express  the  ratio  of  54  ft.  to  72  ft.  in  terms  of  the 
smallest  numbers  possible. 

The  ratio  of  54  ft.  to  72  ft.=  —  =  —  =  -,  dividing  both  terms  of  the  ratio 
72      12     4'  ^ 

by  6  and  then  by  3. 

117 


118  ARITHMETIC 

Exercise  72 

Express  the  ratios  of  the  following  quantities  in  terms  of  the 
smallest  numbers  possible : 

1.  Of  16  ft.  to  48  ft.  5.  Of  54  bu.  to  81  bu. 

2.  Of  $  60  to  1 90.  6.  Of  96  yr.  to  144  yr. 

3.  Of  35  mi.  to  42  mi.'  7.  Of  210  T.  to  126  T. 

4.  Of  42  mi.  to  106  mi.  8.  Of  240  T.  to  180  T. 

Find  the  ratio  of : 

9.   20  to  24.  13.  90  to  108.  17.  36  to  60. 

10.  45  to  60.  14.  64  to  24.  -    18.  64  to  80. 

11.  m  to  63..  15.  125  to  75.  19.  189  to  168. 

12.  50  to  75.  16.  168  to  210.  20.  42  to  182. 

21.  Find  all  the  measures  that  can  be  used  to  measure  the 
capacity  of  each  of  two  baskets  containing  20  qt.  and  32  qt. 

22.  Find  the  lengths  of  the  two  longest  boards  that  can  be  used 
to  build  a  fence  around  a  garden  30  ft.  long  and  24  ft.  wide. 

'  23.  A  farmer  has  66  bu.  of  corn  and  90  bu.  of  wheat,  which  he 
wishes  to  put  into  sacks  of  equal  size,  without  mixing  the  two 
kinds  of  grain.  How  many  bushels  must  each  sack  contain  in 
order  to  be  as  large  as  possible  ? 

PRIME  NUMBERS 

75.  A  Prime  Number  is  one  that  can  be  divided  only  by 
unity  and  itself,  as  5,  11,  and  18. 

Select  the  prime  numbers :  2,  3,  4,  5,  6,  7,  8,  9,  10,  11. 

The  prime  factors  of  a  number  are  the  prime  numbers 
which  when  multiplied  together  give  it;  thus,  3,  3,  and  5 
are  the  prime  factors  of  45. 

4  is  the  second  power  of  2,  and  8  the  third  power.  4  =  2^  ; 
8  =  2x2x2  =  23;  9  =  32 ;  27  =  3  x  3  x  3  =  S^;  16=2*. 


GREATEST   COMMON  MEASURE  119 


76.   Find  the  prime  factors  of  168. 

168 


21 


7 
.-.  168  =  8  X  3  X  7  =  23  X  3  X  7. 

A  number  is  exactly  divisible  : 

By  2  if  its  right-hand  figure  is  an  even  number  or  zero. 
By  5  if  its  right-hand  figure  is  0  or  5. 
By  10  if  the  right-hand  figure  is  0. 

Select  the  numbers  divisible  by  2  or  5  or  10 :  36,  24,  17,  40,  64,  85,  90, 
180,  245. 

Exercise  73 

1.  Name  the  even  numbers  from  1  to  50. 

2.  Name  the  odd  numbers  from  1  to  51. 

3.  Name  the  prime  numbers  from  1  to  100  and  commit  them 
to  memory. 

Eind  the  prime  factors  of : 

4.  30;  36;  56;  48;  84;  66-,  196;  195;  231. 

5.  86;  147;  104;  132;  78;  135;  342;  255. 

6.  336 ;  408 ;  372 ;  564 ;  342 ;  484 ;  375  ;  861. 

7.  What  prime  factors  are  common  to  30  and  36  ?  66  and 
132?     147  and  336?     135  and  255  ? 

8.  Find  the  prime  factors  of  all  numbers  from  1  to  100. 

77.  (1)  A  man  owns  a  rectangular  lot  210  ft.  long  and 
144  ft.  wide.  Find  the  length  of  the  longest  board  that 
can  be  used  to  fence  it. 

We  are  required  to  find  the  length  of  the  longest  board,  i.e.  the  G.  C.  M. 
of  144  ft.  and  210  ft. 

144  =  9  X  16  =  3-^  X  2*. 

210  =  10x21  =  2x5x3x7. 


120  ARITHMETIC 

Thus,  the  G.  C.  M.  of  144  and  210  =  2  x  3  =  6. 
.'.  the  length  of  the  longest  board  is  ^  ft. 
To  prove  the  answer  correct ; 

The  number  of  boards  required  for  the  length  =  210  h-  6  =  36. 
The  number  of  boards  required  for  the  width  =  144  -h  6  =  24. 

35  and  24  have  no  common  measure  except  unity,  /.  6  ft.  is  the  correct 
answer. 

(2)  A  certain  school  consists  of  132  pupils  in  the  high 
school,  154  in  the  grammar,  and  198  in  the  primary  grades. 
If  each  group  is  divided  into  sections  of  the  same  number 
containing  as  many  pupils  as  possible,  how  many  pupils  will 
there  be  in  each  section  ? 

We  are  required  to  find  the  number  of  pupils  in  each  section,  i.e.  the 
G.  C.  M.  of  132,  154,  and  198  pupils. 


2 

132 

154 

198 

11 

66 

77 

99 

6 

7 

9 

Since  2  and  11  are  the  only  common  factors,  the  G.  C.  M.  of  132,  154,  and 
198  is  2  X  11,  or  22. 

.'.  each  section  will  contain  22  pupils. 

Exercise  73(a) 

1.  Draw  two  lines,  one  15  in.  and  the  other  21  in.  long.  What 
is  the  longest  line  that  can  be  used  to  measure  both  lines  ? 

2.  What  is  the  longest  line  that  will  exactly  measure  two 
lines  28  and  32  in.  long  ? 

3.  What  is  the  longest  line  that  will  exactly  measure  three 
lines  respectively  20,  30,  and  45  in.  long  ? 

4.  What  is  the  largest  unit  of  capacity  that  can  be  used  to 
measure  the  quantity  of  oil  in  each  of  two  vessels,  one  containing 
16  qt.  and  the  other  36  qt.  ? 

5.  What  is  the  largest  unit  of  money  that  can  be  used  to  pay 
each  of  two  debts,  one  of  $  45  and  the  other  of  $  80  ? 


LEAST   COMMON   MULTIPLE  121 

6.  Find  the  G.  C.  M.  of  24  and  42 ;  30  and  45 ;  36  and  54. 

7.  Prove  that  your  answer  in  each  case,  in  example  6,  is  a  com- 
mon factor  by  dividing  it  into  each  of  the  numbers.  Prove  that 
it  is  the  greatest  common  measure  by  examining  your  quotients 
and  finding  that  they  have  no  common  measure  except  unity. 

8.  State  how  to  find  the  G.  C.  M.  of  two  numbers. 

Find  the  G.  C.  M.  of: 

9.  40,56.  17.  210,455. 

10.  42,  54.  18.  287,  369. 

11.  81,  105.  19.  230,  506. 

12.  108,  162.  20.  42,  72,  180. 

13.  63,  91.  21.  60,  135,  165. 

14.  90,  105.  22.  210,  462,  546. 

15.  102,  114.  23.  395,  474,  632. 

16.  75,  175.  24.  666,  738,  954. 

LEAST  COMMON  MULTIPLE 

78.  The  quantity  15  in.  is  measured  by  the  unit  5  in.,"  3 
times,  and  is  therefore  called  a  multiple  of  5  in. 

The  quantity  18  lb.  is  exactly  divisible  by  the  units  1  lb., 
2  lb.,  3  lb.,  6  lb.,  and  9  lb.,  and  is  a  multiple  of  each  one  of 
them.  Thus  18  lb.  is  equal  to  18(1  lb.),  9(2  lb.),  6(3  lb.), 
or  3(6  lb.). 

Select  from  the  following  quantities  the  multiples  of  the 
unit  83 :  $12,  116,  |18,  $25,  and  127.  Name  all  the  units 
that  will  exactly  measure  the  quantity  24  hr. 

Any  quantity  is  a  multiple  of  a  unit  of  measure  when  it  is 
exactly  divisible  by  the  unit. 

79.  Thirty  days  is  exactly  divisible  by  the  units  3  da.  and 
5  da.,  and  is,  therefore,  a  common  multiple  of  3  da.  and  5  da. 


122  ARITHMETIC 

One  quantity  is  a  common  multiple  of  two  or  more  units 
when  the  former  is  exactly  divisible  by  each  of  the  latter. 

Thirty  days  is  the  least  quantity  that  is  exactly  divisible 
by  the  units  6  da.  and  10  da.,  and  is,  therefore,  the  least  com- 
mon multiple  of  6  da.  and  10  da. 

The  Least  Common  Multiple  (L.C.M.)  of  two  or  more  units 
is  the  least  quantity  that  is  exactly  divisible  by  each  of  them. 

For  convenience,  we  speak  of  one  number  being  a  multiple 
of  another,  or  a  common  multiple,  or  the  least  common  mul- 
tiple of  two  or  more  numbers. 

80.  The  smallest  number  in  the  multiplication  table  of  6 
and  also  of  9  is  18.     Therefore  18  is  the  L.  C.  M.  of  6  and  9. 

Exercise  74 
Find  by  the  method  of  the  preceding  paragraph  the  L.  C.  M.  of : 

1.  4,  5.  4.   9,  12.  7.   2,  3,  6.  10.   5,  4,  10. 

2.  6,  7.  5.    10,  12.  8.    2,  3,  4.  11.   6,  8,  12. 

3.  6,  8.  6.   8,  12.  9.    4,  5,  6.  12.   3,  6,  9. 

13.  Find  the  L.  C.  M.  of  the  denominators  of  these  fractions : 

%i  hi  I.  i.  I 


i>l 

i,  f 

\,hi 

A.  f 

A.  « 

4.  *.  H 

?>  1 

i,  A 

1.  f,  i 

81.    (1)  Find  the  shortest  distance  which  can  be  exactly 
measured  by  two  lines  respectively  36  ft.  and  48  ft.  long. 

We  are  here  required  to  find  the  shortest  distance,  i.e.  the  L.  C.  M.,  of  the 
units  36  ft.  and  48  ft. 


LEAST   COMMON  MULTIPLE 


123 


36  =  4  X  9 

48  =  3  X  16 


22  X  32. 
3x  2*. 


Thus  the  L.  C.  M.  of  36  and  48  =  2*  x  32  =  16  x  9  =  144. 
.-.  the  shortest  distance  is  144  ft. 

(2)  Find  the  L.C.M.  of  24,  30,  36. 

24    30    36 


6 


2      5      3 

Here  6  and  2  are  the  factors  common  to  two  or  more  of  the  numbers,  and 
2,  5,  and  3  are  the  factors  not  common. 

.  •.  the  L.  C.  M.  =  6  X  2  X  2  X  5  X  3  =  360. 

State  how  to  find  the  L.  C.  M.  of  two  or  more  numbers. 

Show  by  division  that  24,  30,  and  36  are  all  factors  of  their  L.  C.  M.  360. 

(3)  Find  the  L.  C.  M.  of  14,  21,  54,  bQ,  84. 


3 

X^    ^X    54    56    84 

2 

18     56     ^^ 

9    28 

.-.  the  L.  C.  M.  =  3  X  2  X  9  X  28  =  1512. 

14  is  erased  since  it  is  a  factor  of  56,  and  21  since  it  is  a  factor  of  84.    In 
the  second  line,  28  is  erased  since  it  is  a  factor  of  56. 


Exercise  75 
Find  the  L.  C.  M.  of: 

1.  4,  8,  16,  32.  4.   15,  18,  28,  36.  7.   65,  26,  56,  52. 

2.  3,  6,  9,  12.  5.   12,  20,  21,  45.  8.   36,  48,  60,  54. 

3.  24,  30,  36,  45.            6.   22,  33,  30,  44.  9.   33,  27,  55,  135. 
10.   30,  21,  40,  28,  24,  56.                 11.  56,  36,  63,  28,  72. 

Find  the  L.  C.  M.  of  the  denominators  of  these  fractions : 
12.    i,  i?  :f'                        13.    2-5-,  ^,   Q-Q.  14.    ^,  Tjr,  Y^,  Y^. 

1^-     "??    t'    2^T>    TT*  l''^-  "S?    T5f    f2"'    T^' 

16-    h  h  h  «•  18.  i,  f,  ^,  H. 


CHAPTER   XI 

FRACTIONS 

82.  If,  on  measuring  a  certain  quantity  with  a  unit  1  ft. 
long,  I  count  6  units  in  the  quantity,  then  I  know  that  the 
quantity  is  6  times  1  ft.,  or  6  ft.  Similarly,  if  I  measure 
another  quantity  with  a  unit  2  ft.  long  and  count  4  units  in 
the  quantity,  I  know  that  it  is  4  times  2  ft.,  or  4  (2  ft.). 
If  I  measure  a  third  quantity  with  a  unit  J  ft.  long  and 
count  3  units  in  the  quantity,  then  I  know  that  it  is 
3  times  the  unit  J  ft.,  or  |  ft.  Thus  when  a  measured 
quantity  is  denoted  by  the  expression  |  ft.,  the  measuring 
unit  is  I  ft.  and  the  number  of  units  in  the  quantity  is  3. 

Let  actual  work  he  done  by  the  class  in  measuring  unknown 
quantities  with  fractional  units. 

Exercise  76 

1.  I  measured  a  quantity  with  the  unit  J  ft.  and  counted  2 
units.     What  was  the  quantity  ? 

2.  Name  the  quantities  which,  when  measured  by  the  following 
units,  give  the  numbers  indicated : 


Unit 

Number 

Unit 

Number 

ift. 

5 

|da. 

5 

iVft. 

7 

igal. 

7 

i  mi. 

5 

^^  doz. 

18 

i,\h. 

13 

ibu. 

7 

ilb. 

9 

AT. 

9 

ihr. 

4 

i^A. 

11 

124 


FRACTIONS  125 

83.  The  expression  f  ft.  denotes  that  the  quantity  1  ft. 
is  conceived  as  made  up  of  6  equal  parts  or  units,  and  that 
5  of  these  parts  or  units  have  been  taken  to  measure  the 
quantity  denoted  by  |  ft. 

The  primary  unit,  1  ft.,  has  been  divided  into  6  equal 
parts  to  give  the  measuring  unit,  which  is  ^  ft.,  or  2  in. 
The  number  of  these  units  in  the  given  quantity  is  5.  The 
ratio  of  the  given  quantity  to  the  measuring  unit  is  5. 

84.  The  quantity  represented  by  |  ft.  contains  5  direct 
measuring  units,  and  the  primary  unit,  1  ft.,  contains  6  of 
these  units.  Henee^  the  fraction  J  expresses  the  ratio  of  the 
quantity  denoted  hy  |  ft.  to  the  primary  unit,  1  ft. 

c[ ! I [ [ 12) 

^I \ I 1 I I ]B 

85.  Draw  a  line,  AB,  1  ft.  long.  Divide  it  into  6  equal  parts,  or  units, 
each  ^  of  a  foot  long.  Draw  a  second  line,  CD,  above  the  first,  containing 
5  of  these  units,  and  use  these  two  lines  to  illustrate  the  preceding  paragraph. 

Exercise  77 

In  the  following  quantities  name  the  measuring  units,  and  state 
the  number  of  units  in  each  quantity : 


1. 

fft. 

5.    $f. 

9. 

If  sq.  ft. 

13. 

f  da. 

2. 

fyd. 

6.    $|. 

10. 

f  cu.  yd. 

14. 

fda. 

3. 

Jib. 

7.    $tV 

11. 

f  wk. 

15. 

Hhr. 

4. 

|sq. 

yd. 

8.    $ii' 

12. 

Hj^- 

16. 

f  min. 

17.  Make  a  drawing  to  show  that  the  ratio  of  |  ft.  to  1  ft.  is  f . 
Make  a  drawing  to  show  that  the  ratio  of  |  yd.  to  1  yd.  is  f. 

18.  Give  the  ratio  of  each  quantity,  in  examples  1  to  16,  to  its 
primary  unit. 


126  ARITHMETIC 

86.  A  Fraction  is  a  number  in  which  the  unit  of  measure 
is  a  definite  part  of  some  primary  unit  of  the  same  kind. 

The  denominator  shows  into  how  many  parts  the  primary 
unit  is-  divided  to  give  the  unit  of  measure ;  it  also  names 
this  unit.  The  numerator  shows  the  number  of  them  that 
measures  the  quantity. 

A  proper  fraction,  as  an  expression  of  measured  quantity, 
is  one  in  which  the  numerator  is  less  than  the  denominator. 
Select  the  proper  fractions :  |,  ^,  -|,  |,  |. 

A L \ ! I \b 

d\  \  \ [ I I \e 

87.  Let  AB  represent  some  quantity  measured  by  5  units, 
each  equal  to  A  (7,  and  J)E  as  measured  by  6  units,  each  equal 
to  A  C.  Then  if  we  think  of  AB  in  relation  to  DE^  we  think 
of  5  units  in  relation  to  6  units,  and  this  relation  or  ratio  is 
expressed  by  the  fraction  |. 

Similarly,  the  fraction,  or  number  |,  expresses  the  ratio  of 
8  5  to  1 6,  5  hr.  to  6  hr.,  5  mi.  to  6  mi.,  5  (8  ft.)  to  6  (8  ft.), 
5  (12  lb.)  to  6  (12  lb.),  or,  generally,  5  of  any  unit  to  6  of 
the  same  unit. 

Similarly,  it  expresses  the  ratio  of  20  lb.  (i.e.  5x4  lb.) 
to  24  lb.  (i.e.  6x4  lb.),  and  so  on. 

Exercise  78 

1.  Name  quantities  whose  ratio  is  |;  f;  f;  |;  ^. 

Write  the  fraction  which  expresses  the  ratio  of  the  following 
quantities : 

2.  $4  to  19;  $8  to  112;  1  dime  to  $1. 

3.  3  qt.  to  4  qt. ;  2  qt.  to  1  gal. 

4.  12  yd.  to  32  yd. ;  4  in.  to  2  ft. 


FRACTIONS 


127 


5.  If  32  yd.  of  cloth  cost  $48,  what  part  of  $48  will  12  yd. 
cost  ?     How  many  dollars  will  12  yd.  cost  ? 

6.  What  is  the  ratio  of  24  men  to  36  men  ?   18  men  to  24  men  ? 

7.  If  18  men  can  do  a  piece  of  work  in  32  da.,  in  what  part 
of  32  da.  can  24  men  do  the  same  work  ?     How  many  days  ? 

88.  Let  AB  represent  some  definitely  measured  quantity, 
as  4  ft.  or  16  ft.,  and  let  it  be  divided  as  shown  in  the 
diagram. 


1 

■■ 

-f 

+         1 

1 

4 

1 

f        1       -f         1 

8 

8 

J- 

8 

1 

8 

1 

8 

8         1         8 

-M 

1         1 

16      10 

16      16 

T^l6_ 

_J_  JL 
16      16 

1        1 
10       16 

16 

.  _i.     J_    J_ 
16       16      10 

16      16    I 

+ 

- 

1 
3 

+           ■ 

i 

i 

i 

i 

i        1 

T^      A 

l2     1     12 

-r 

12 

1 
12 

1      1      1 
T2    1    -12- 

'TtT 

r 

TTi 

T 

5 

5 

■5- 

■"     1 

■h    -k 

10           10 

i     -fe 

1             1 
10            10 

10        10     1 

4     5     6     8     10     1.2 
4'    5'   6'  ^'    10'    12' 

It  is  also 


It  is  evident  from  this  diagram  that  | 
^|,  of  a  quantity  measure  it,  and  are  all  equal, 
evident  that  |,  |,  |,  |,  -^q,  -f^^  ^^g,  of  a  quantity  measure  one- 
half,  of  it,  and  are  all  equal. 

Similarly,  J,  |,  ^^2^  of  a  quantity  measure  one-third  of  it, 
and  are  equal.      Similarly,  |-  —  yo- 

89.  The  fraction  -f^  is  got  from  the  fraction  |  by  multiply- 
ing both  numerator  and  denominator  by  6.  -^q  reduces  to  ^ 
by  dividing  both  numerator  and  denominator  by  8.     How 


128  ARITHMETIC 

can  you  change   the  fraction   |  to  j^^?   ^  to  ^^^7   |  to  J? 

Multiplying  or  dividing  both  terms  of  a  fraction  by  the  same 
number  does  not  change  its  value. 

Exercise  79 

1.  By  what  must  you  multiply  both  numerator  and  denomi- 
nator of  I  to  get  f  ?  By  what  must  you  divide  both  numerator 
and  denominator  of  -f  to  reduce  it  to  |  ? 

2.  Make  a  drawing  to  show  that  |  =  f . 

Use  the  results  you  obtain  in  the  following  to  verify  the  rule 
in  the  preceding  paragraph : 

3.  Find!  of  $8;  fof  ^8;  f  of  $8. 

4.  Find  ^  of  $30;  j\  of  $30;  j\  of  $30. 

5.  Find  ^  of  36  ft. ;  find  also  respectively  |^,  J,  -j^,  3^,  and  ^ 
of  36  feet. 

6.  Find  |  of  24  dimes  ;  |  of  24  dimes ;  y\  of  24  dimes. 

7.  Find  f  of  28  lb. ;  |J  of  28  lb. ;  ff  of  28  lb. 

8.  Find  f  of  32  da. ;  f  of  32  da. ;  H  o^  ^2  da. ;  U  oi  32  da. 

9.  I  da.  =  ?  hr. ;  j\  da.  =  ?  hr. ;  |f  da.  =  ?  hr. ;  |  da.  =  ?  hr. 

10.  f  hr.  =  ?  min. ;  J-|  hr.  =  ?  min. ;  ^^  hr.  =  ?  min. ;  }  hr.  =  ?  min. 

11.  What  actual  coins  and  how  many  of  each  kind  are  equal 
respectively  to  |,  |,  y%,  and  ^^  of  $  1  ? 

90.    (1)  Express  9  yd.  as  eighths  of  a  yard. 

A  \  \  \  \  \  \  \  B    . 

Let  the  line  AB  be  drawn  to  represent  1  yd.  Think  of  1  yd.  as  contain- 
ing 8  units  of  length,  as  shown  in  the  diagram. 

Then  9  yd.  will  contain  9  x  8,  or  72  of  these  units. 
Therefore  9  yd.  is  equal  to  ^^  of  1  yd. 


FRACTIONS  129 

(2)  Express  f  |  as  a  fraction  with  20  as  a  denominator. 

Think  of  $  1  as  containing  20  units  of  5  ^  each.  Then  $  |  contains  f  of 
20,  or  15  of  these  units.    Therefore,  $|  =  ^^f. 

Exercise  80 
Express  as  fractions : 

1.  $8asl0tlis.  7.  5  yr.  as  12tlis. 

2.  $4  as  20ths.  8.  3  da.  as  24tlis. 

3.  9  yd.  as  3ds.  9.  4  m  in.  as  60ths. 

4.  6  ft.  as  12ths.  10.  6  hr.  as  60ths. 

5.  14  gal.  as  4ths.  11.  8  pk.  as  Sths. 

6.  11  wk.  as  7ths.  12.  12  bu.  as  4ths. 

13.  In  each  of  these  questions  name  the  measuring  unit  in 
your  fraction  as  an  actual  unit  of  measure  in  common  use. 

14.  Express  as  fractions  with  100  as  denominator :   -J,  ^,  ^,  ji^, 

1         13      3      2       9 

Y¥>  2Z}  ¥'  "5^'  o"'  "nr* 

Reduce,  illustrating  your  work  by  diagrams : 

15.  f  ft.  to  12ths.  19.    I  da.  to  24ths. 

16.  I  yd.  to  33ds.  20.    ^%  to  78ths. 

17.  I  lb.  to  16ths.  21.    -1%  to  45ths. 

18.  f  gal.  to  28ths.  22.    ii  to  54ths. 

23.  What  are  the  new  units  of  measure  in  your  results  ? 
Where  possible,  identify  them  with  actual  units  in  common  use. 

91.    (1)  Express  8  in.  as  a  fraction  of  1  ft. 
8  in.  =  ^j  or  I  of  a  foot  =  f  ft. 

(2)  A  man's  capital  is  represented  by  20  units  of  value. 
He  invests  ^  of  it  in  land  and  1  of  the  remainder  in  bank 
stock.  How  many  units  did  he  invest  in  bank  stock,  and 
what  part  is  it  of  his  entire  capital  ? 


130  ARITHMETIC 

The  amount  invested  in  land  =  :|  of  20  units,  or  5  units. 

The  remainder  =  20—5,  or  15  units. 

The  amount  invested  in  stock  =  |  of  15,  or  5  units. 

.  •.  the  amount  invested  in  stock  =  ^^,  or  ^  of  his  entire  capital. 

Exercise  81 

1.  Express  as  a  fraction  of  a  foot: 

3  in.,  4  in.,  5  in.,  6  in.,  9  in.,  10  in.,  15  in.,  16  in.,  27  in. 

2.  At  8)^  a  foot  what  is  the  cost  of  9  in.  of  rubber  tubing? 
Of  18  in.  ?     Of  30  in.  ? 

3.  Express  as  a  fraction  of  a  pound  Avoirdupois: 

14  oz.,  6  oz.,  9  oz.,  12  oz.,  15  oz.,  20  oz.,  24  oz. 

4.  At  20^  a  pound  what  is  the  cost  of  12  oz,  of  cheese? 
20  oz.  ?     24  oz.  ? 

5.  Express  as  a  fraction  of  a  month : 

12  da.,  15  da.,  18  da.,  20  da.,  25  da.,  40  da.,  45  da.,  54  da. 

6.  A  commercial  traveller  was  away  from  home  27  da.  in  the 
month  of  April.     What  part  of  the  month  was  he  at  home  ? 

7.  If  1  mi.  is  the  measuring  unit,  what  number  measures 
each  of  the  following: 

640  rd.?     20  rd.?    35  rd.  ?     80  rd.  ?     120  rd.  ?     150  rd.  ? 

8.  If  a  bicyclist  rides  1  mi.  in  8  min.,  how  long  does  it  take 
him  to  ride  60  rd.  ? 

9.  Express  as  a  fraction  of  a  day : 

6  hr.,  9  hr.,  12  hr.,  15  hr.,  18  hr.,  24  hr.,  32  hr.,  36  hr. 

10.  If  a  baby  sleeps  16  hr.  a  day,  what  part  of  the  day  is  he 
awake  ? 

11.  Express  as  fractions  of  a  dollar: 

25,  50,  75,  60,  80,  100,  125,  150,  and  160  cents. 

12.  At  $1  a  yard,  how  much  cloth  can  I  buy  for  $2.50  ? 


FRACTIONS  131 

13.  If  $4  is  used  us  a  measure  of  $4,  what  is  the  number  ex- 
pressing the  measurement  ?  If  $  4  is  used  as  a  measure  of  $  3, 
what  is  the  number? 

14.  If  cloth  is  40^  a  yard,  how  much  can  I  buy  for  30^  ? 

15.  If  oranges  cost  30^  a  dozen,  what  part  of  a  dozen  can  I 
buy  for  25^?     How  many  oranges? 

16.  A  man  walks  a  certain  distance  in  4  hr.  What  part  of  it 
does  he  walk  in  1  hr.  ?     In  2  hr.  ?     In  |  hr.  ?     In  i  hr.  ? 

17.  A  can  do  a  piece  of  work  in  3  hr.,  B  the  same  amount  in 
4  hr.  If  the  work  be  measured  by  12  units,  how  many  units  will 
A  do  in  an  hour  ?  How  many  B  ?  How  many  both  working 
together  ?  What  part  will  their  joint  work  for  an  hour  be  of  the 
whole  work? 

18.  A.  and  B  put  $8000  into  a  business.  B  put  three  times  as 
much  as  A.  How  many  units  measure  is  A's  share  ?  B's  share  ? 
The  entire  amount  ?  A  put  in  what  part  of  $  8000  ?  How  much  ? 
B  what  part  ?     How  much  ? 

19.  A  received  a  certain  sum  of  money,  B  twice  as  much,  and 
C  as  much  as  A  and  B  together.  How  many  units  measure  the 
entire  amount  ?    What  part  of  the  whole  sum  does  each  receive  ? 

20.  If,  in  the  previous  example,  $24  was  divided  among  A,  B, 
and  C,  find  the  share  of  each. 

21.  Given  that  pure  water  contains  15  parts  by  weight  of 
oxygen,  and  2  parts  of  hydrogen,  what  part  of  the  weight  of  a 
gallon  of  water  is  hydrogen  ? 

22.  How  many  pounds  of  oxygen  and  of  hydrogen  in  34  lb.  of 
water  ? 

23.  Six  brothers  join  in  paying  a  debt  of  $700.  The  eldest 
pays  ^  of  it,  and  each  of  the  others  -J-  of  the  remainder.  How 
much  does  the  eldest  pay  ?  How  much  does  each  of  the  five 
pay  ?     This  is  what  part  of  the  whole  debt  ? 

24.  If  a  pipe  empties  a  tank  at  the  rate  of  12  gal.  in  1  min., 
what  is  the  rate  per  second? 


132  ARITHMETIC 

25.  $40  is  divided  among  A,  B,  and  C,  giving  A  f  of  it,  B  |, 
and  C  the  remainder.  What  is  the  sum  of  A's  and  B's  shares  ? 
What  is  C's  share  ?     C's  share  is  what  part  of  the  whole  sum  ? 

26.  The  value  of  a  mine  is  represented  by  10  units  of  money. 
A  man  who  owns  |  of  it  sells  f  of  his  share.  How  many  units 
did  he  own  ?  How  many  did  he  sell  ?  What  part  of  the  whole 
mine  did  he  sell  ? 

27.  The  length  of  an  oblong  is  8  ft.,  and  the  width  4  ft. 
What  part  of  the  perimeter  is  the  length? 

28.  An  oblong  6  ft.  wide  'and  8  ft.  long  is  divided  into  strips 
each  1  ft.  wide,  made  by  drawing  lines  parallel  to  the  length. 
What  part  of  the  area  of  the  oblong  is  the  area  of  one  strip  ? 

29.  The  area  of  an  oblong  is  16  sq.  ft.  What  part  of  its  area 
is  that  of  a  square  whose  side  is  2  ft.  ? 

30.  If  20  units  measure  the  cost  of  a  pound  of  tea  sold  at  a 
gain  of  -^-^  of  the  cost,  how  many  units  are  gained  by  selling  ? 
What  is  the  selling  price?  The  cost  price  is  what  part  of  the 
selling  price  ?  What  is  the  ratio  of  the  selling  price  to  the  cost  ? 
The  gain  is  what  part  of  the  selling  price  ? 

31.  A  has  12  marbles,  and  B  has  3.  They  play  together,  and 
A  loses  ^  of  his  marbles.  How  many  has  B  now  ?  What  part 
are  they  of  what  A  now  has? 

32.  The  value  of  a  house  is  measured  by  5  units.  The  lot  on 
which  it  stands  is  worth  ^  as  much  as  the  house.  What  is  the 
measure  of  the  value  of  the  house  and  lot  ?  The  value  of  the  lot 
is  worth  what  part  of  both  together  ? 

33.  A  and  B  set  out  at  the  same  time  from  places  42  mi.  apart, 
and  meet  at  the  end  of  6  hr.  A  travels  at  the  rate  of  3  mi.  an 
hour.  How  far  does  B  travel  ?  What  is  B's  rate  ?  A's  rate  is 
what  part  of  B's  ? 

34.  Apples  are  sold  at  the  rate  of  12  for  a  dime,  and  bananas 
at  the  rate  of  8  for  a  dime.  Compare  the  value  of  an  apple  with 
that  of  a  banana. 

NoTB.  —  Let  the  dime  be  measured  by  24  units. 


REDUCTION  OF  FRACTIONS  183 

35.  A  crew  can  row  6  mi.  an  hour  in  still  water.  What  is  the 
rate  of  rowing  up  stream  in  a  current  running  at  the  rate  of  2  mi. 
an  hour?  What  is  the  rate  down  stream?  What  is  the  ratio 
of  the  rate  down  stream  to  that  up  stream  ? 

REDUCTION   OF   FRACTIONS 

92.  A  fraction  is  in  its  lowest  terms  when  its  numerator 
and  denominator  have  no  common  factor. 

(1)  Reduce  1 1^|  to  its  lowest  terms. 

The  object  of  reduction  is  to  give  a  more  definite  idea  of  the  value  of  the 
quantity  by  expressing  the  ratio  in  the  smallest  numbers. 

The  common  factors  are  3,  3,  and  5. 

The  effect  of  dividing  each  term  of  the  first  fraction  by  3  is  to  make  each 
measuring  unit  3  times  as  large,  and  to  reduce  the  number  of  these  units  to 
one-third  as  many.  Similarly,  with  the  second  division  by  3,  and  the  third 
by  5. 

(2)  A  merchant  paid  66  ^  a  yard  for  cloth  and  sold  it  for 
88  ^  a  yard.    What  fraction  of  the  cost  was  the  selling  price  ? 

The  selling  price  =  f  |  or  f  of  the  cost. 

Exercise  82 
Reduce  to  its  lowest  terms  : 


1. 

«H- 

4. 

fit- 

7. 

u- 

10. 

H- 

is. 

n- 

2. 

Hi- 

5. 

fu- 

8. 

u- 

11. 

fi- 

14. 

AV 

3. 

«M- 

6. 

ll- 

9. 

a- 

12. 

n- 

15. If  I  pay  $60  for  a  bicycle  and  afterward  sell  it  for  f  45, 
what  part  of  the  cost  do  I  sell  it  for  ? 

16.  A  merchant  sells  55  yd.  of  cloth  from  a  piece  containing 
66  yd.     What  part  of  the  whole  piece  does  he  sell  ? 


134  ARITHMETIC 

17.  A  man  buys  a  horse  for  $  128  and  sells  it  for  $  112.  Find 
the  selling  price  as  a  fraction  of  the  cost. 

18.  On  an  investment  of  $88  a  merchant  gains  $33.  Find  the 
ratio  of  the  gain  to  the  cost. 

19.  Sixty  days  is  what  fraction  of  a  year  ? 

20.  Out  of  a  farm  containing  135  A.,  81  A.  were  sold.  Find 
what  part  of  the  farm  was  sold. 

21.  1  ft.  8  in.  =  ?  in.  2  ft.  6  in.  =  ?  in.  What  is  the  ratio  of 
1  ft.  8  in.  to  2  ft.- 6  in.? 

22.  What  is  the  ratio  of  2  ft.  9  in.  to  3  ft.  8  in.  ?  Of  6  gal.  1  qt. 
to  7  gal.  2  qt.  ?  Of  2  lb.  8  oz.  to  3  lb.  2  oz.  ?  Of  2  bu.  1  qt.  to 
3  bu.  8  qt.  ? 

23.  If  2  lb.  3  oz.  of  cheese  cost  30^,  what  will  1  lb.  5  oz.  cost? 

24.  If  3  doz.  6  eggs  cost  $  .50,  what  will  5  doz.  3  eggs  cost  ? 

93.  Make  drawings  to  show  the  difference  between  2|  ft. 
and  ^^-  ft.  In  your  first  drawing  how  many  units  each  equal 
to  1  ft.  did  you  mark  off?  How  many  equal  to  ^  ft.  ?  Can 
you  count  5  units  in  this  quantity  ?  Why  not  ?  In  your 
second  drawing  how  many  units,  each  ^  ft.,  can  you  count? 

94.  Such  an  expression  (as  2|  ft.,  e.g.^  denotes  a  quantity 
in  which  two  units  of  measure  of  different  values  have  been 
used  :  a  primary  unit  and  parts  of  this,  a  derived  unit. 

2|  is  the  number  11  in  disguise.  To  make  it  number  in  the 
strict  sense,  we  must  express  the  quantity  in  the  smaller  unit 
of  measure,  i.e.  as  ^  ft. ;  this  as  a  quantity  can  be  counted; 
2 1  cannot  be  counted. 

An  improper  fraction  as  an  expression  of  measured  quan- 
tity is  one  whose  numerator  is  equal  to  or  greater  than  its 
denominator  ;  as  |,  f,  |,  ^. 


REDUCTION  OF  FRACTIONS  135 

95.   Reduce  to  an  improper  fraction  5|  yd. 
Let  1  yd.  =    3  units  of  ^  yd.  each. 

Then  5  yd.  =  15  units  of  i  yd.  each. 

I  yd.  =    2  units  of  i  yd.  each. 
.-.  5|  yd.  =  17  units  of  |  yd.  each,  or  -J/  yd. 

Hence,  if  we  divide  the  quantity  5f  yd.  into  parts,  each  equal  to  |  yd. , 
we  can  count  17  parts  in  the  quantity. 

Thus,  we  may  multiply  3  by  5  and  add  2,  or,  by  the  law  of  commutation, 
multiply  5  by  3  and  add  2  to  get  the  numerator. 

Exercise  83 
Reduce  to  improper  fractions  : 

1.  2\  ft.,  2f  ft.,  31  ft.,  5f  ft.,  6i  ft. 

In  this  example  by  "what  number  do  you  multiply  each  time  ? 

2.  3|  yd,  4f  yd,  3|  yd,  7^  yd,  8f  yd. 
What  is  your  multiplier  here  ? 


3. 

^41 

8. 

8iyd. 

13. 

12f 

18. 

20f. 

4. 

$4f. 

9. 

5 A  ft. 

14. 

9|. 

19. 

21f. 

5. 

$7f. 

10. 

20f  gal. 

15. 

12A. 

20. 

19f. 

6. 

^161 

11. 

8|pk. 

16. 

6tV 

21. 

7t%. 

7. 

^9f^. 

12. 

5f\  lb. 

17. 

3tV. 

22. 

40J. 

23.  What  are  the  two  units  of  measure  in  example  8?  What 
is  the  unit  of  measure  in  the  result  ?  How  many  of  these  units 
must  be  counted  to  measure  the  quantity  ? 

24.  What  is  the  quantity  which  is  measured  by  the  unit  |  ft. 
and  the  number  9  ?  If  the  same  quantity  be  measured  by  the 
number  3,  what  is  the  measuring  unit  ? 

25.  If  9  boys  receive  f  4  each,  what  sum  was  divided?  If 
the  same  sum  had  been  divided  among  4  boys,  what  would  each 
have  received  ? 


136  ARITHMETIC 

Exercise  84 
Keduce  to  improper  fractions : 


1. 

15^. 

5. 

34A. 

9. 

64if 

13. 

29AV 

2. 

21A. 

6. 

48ii. 

10. 

152|. 

14. 

87M. 

3. 

29|f 

7. 

51M. 

11. 

98A. 

15. 

341«. 

4. 

4l3rV 

8. 

36M. 

12. 

168f. 

16. 

453fH. 

96.  (1)  Which  is  greater,  f  ft.  or  f  ft.  ? 

Consider  the  feet  as  having  42  measuring  units,  then  f  of  this  is  35,  and 
f  is  36  units. 

.'.  ^  ft.  is  greater  than  |  ft.  by  1  unit,  or  ^j  ft. 

(2)  Compare  the  quantities  $  |,  $  3^,  f  |. 

Let  $  1  be  represented  by  24  units  of  value.  Then  $  f ,  $  ^1^,  $  f ,  are  re- 
spectively equal  to  18,  14,  and  15  units.  Hence  the  first  fraction  is  the  great- 
est and  the  second  the  least. 

State  how  to  find  the  greatest  and  least  of  two  fractions. 

Exercise  85 
Find  the  greatest  and  least  of  the  following : 

1.     fyd.,     3   yd.  4.     I  ft.,    fft.,    Hft- 

3.    $fV  ^h  f  A-  6.    I,  I,  tt- 

7.  State  how  to  find  the  greatest  and  least  of  a  number  of 
fractions. 

97.  Express  ^^  yd.  in  terms  of  the  primary  and  derived 
units  of  measure. 

The  primary  unit  1  yd.  contains  5  derived  units  of  ^  of  a  yard  each. 
Hence  ^  yd.,  i.e.  23  derived  units  =  4  primary  and  3  derived  units; 

=  4i  yd. 
Or,  more  simply,  5)23 


KEDUCTION  OF  FRACTIONS  137 

Exercise  86 

Express  in  terms  of  the  primary  and  direct  measuring  units : 

1.  I    ft.  7.        $^^-.  13.       -2^4    y(i. 

'  14.    ^-Jf  rd. 
15.    %8^5_  mi. 


2. 

-V-  ft. 

3. 

¥  yd. 

4. 

¥  yd. 

5. 

^o_  mi. 

6. 

i^mi. 

7. 

^¥- 

8. 

$-«/. 

9. 

11^  da. 

10. 

i|A  wk. 

11. 

-¥2^  yr. 

12. 

-W  ft- 

16.  ^^  oz. 

17.  -\Wlb. 

18.  ^o^Ub. 

19.  What  are  the  two  units  of  measure  in  the  answers  to 
examples  1,  3,  and  5  ? 

Exercise  87 

Reduce  to  integers  and  proper  fractions : 

98.  Reduce  |  ft.,  |  ft.,  |^  ft.  to  fractions  having  for  their 
common  denominator  the  least  common  denominator  (L.C.D.) 
of  these  fractions. 

The  L.  C.  D.  of  3,  4,  and  6  is  evidently  12  ;  the  foot  is  considered  as 
divided  into  12  equal  parts. 

I  ft.  =  8  parts  =  ^  ft. ;  f  f t.  =  9  parts  =  ^^  ft. ;  f  ft.  =  10  parts  =  \%  ft. 
Make  a  drawing  to  show  that  this  result  is  correct. 

Exercise  88 

1.  Reduce  $  i  and  1 1  to  tenths. 

2.  Reduce  $1,  l|,  and  ^^^  to  thirtieths. 

3.  Reduce  |^  yd.,  f  yd.,  and  |  yd.  to  eighteenths. 


138  ARITHMETIC 

Keduce  to  their  least  common  denominator : 

4.  I  ft.,  J  ft.  10.  i,f  16.  J,  I,  J. 

5.  |hr.,ihr.  11.  |,  f  17.  J,  J,  f . 

6.  I  da.,  3^  da.  12.  ^,  |.  18.  h  h  ^• 

7.  fyr.,|yr.  13.  },  ^.  19.  |,  f,  H- 

8.  if.  14.  A,  3^.  20.  |,|,f 

9.  hi'  15.  |,f  -  21.  f,A,i. 

ADDITION  OF  FRACTIONS 

99.   The  sum  of  3  dimes  and  4  dimes  =  7  dimes. 

The  sum  of  $^-^  and  $-^  =  f-Jj^. 

The  sum  of  5  oz.  and  8  oz.  =  13  oz. 

The  sum  of  j\  lb.  and  ^\  lb.  =  if  lb. 

Here  the  measuring  unit  of  value  is  $  J^,  or  1  dime,  that  of 
weight  y^g  lb.,  or  1  oz. 

Why  cannot  you  add  ^  ft.,  f  ft.,  |  ft.  by  simply  adding 
their  numerators  ? 

(1)  Add  J  ft.,  I  ft.,  I  ft. 
The  L. CD.  =  12. 

i  ft.  =  ^,  ft. ;    §  ft.  =  3^  ft.  ;   I  ft.  =  ^5  ft. 

.-.  the  sum  =  ^  +  ^  "^  ^  ft.  =  ff  ft.  =  1^  ft. 

Prove  the  correctness  of  this  result  by  drawing  a  line  and  measuring  off 
^  ft.,  I  ft.,  and  I  ft.,  and  also  1^|  ft. 

(2)  Find  the  sum  of  f  21f,  |15f,  $13|,  f  8^. 
The  L.  C.  D.  =  72. 

4^    6        9        12      ^  72  *^72      '  ^* 

$21  +  ^15  +  $13+  a8  =  $57. 

.-.  the  sum  =  $  57  +  l$2|J  =  $  67|J. 


ADDITION  OF  FRACTIONS  139 

Exercise  89 
Find  the  sum  of : 

1.    fj,  $i.  3.    tV  oz.,  3^  oz.  5.    fwk.,  fwk. 

2-    -Aft.,  3^  ft.  4.    X3  da.,  11  da.  6.    |J  min.,  g^  min. 

Exercise  90 

In  the  following  exercise,  what  is  the  least  common  denominar 
tor  in  each  example  ? 

Find  the  sum  of : 
1.  ^  1  $ i       2.  $  i,  $  t.  ^  f      3.  I  ft.,  f  ft.      4.1  ft.,  f  ft.,  I  ft. 

Prove  your  answers  to  3  and  4  correct  by  measuring  with  a 
ruler. 

5.  I  lb.,  I  lb.,  if  lb.         6.  1  gal.,  f  gal.         7.  f  bu.,  J  bu.,  H  bu. 

8.  i  yr.,  H  yr.  9.  I  da.,  |  da.,  |  da. 

Prove  your  results  correct  by  reducing  each  fractional  quantity 
to  a  lower  denomination  (as  $^  to  cents,  |  ft.  to  inches,  f  lb.  to 
ounces,  and  so  on)  and  also  your  result.     Then  add  the  integers. 

10.  What  is  the  weight  of  J  lb.  of  tea  and  ^  lb.  of  cheese  ? 

11.  A  piece  of  ribbon  can  be  cut  into  two  parts,  one  |  yd.  long 
and  the  other  J  yd.     Find  its  length. 

Exercise  91 

Find  the  sum  of  the  following  fractional  parts  of  any  unit : 


1. 

hi- 

7. 

H,  n- 

13. 

TT5"?  -h- 

2. 

ll 

8. 

9f ,  31 

14. 

^>h 

3. 

hi-    ■ 

9. 

^,  3{. 

15. 

A.A- 

4. 

hi- 

10. 

^,  4f 

16. 

2A,  6^. 

5. 

h^- 

11. 

9f ,  8f 

17. 

3f ,  2Jj. 

6. 

hi- 

12. 

6i  411. 

18. 

5H,  ^W 

140  ARITHMETIC 

19.  State  how  to  find  the  sum  of  two  or  more  (1)  proper  frac- 
tions, (2)  mixed  numbers. 

20.  Explain  clearly  the  principles  involved  in  finding  the  sum 
of  two  fractions. 

21.  A  boy  had  $6^  in  a  bank  and  put  in  $3|.  How  much 
did  he  then  have  in  the  bank  ? 

22.  A  child's  dress  was  made  from  two  remnants,  one  con- 
taining 2^  yd.,  and  the  other  3J  yd.  How  many  yards  did  the 
dress  contain  ? 

23.  Three  brothers  gathered  walnuts.  The  first  gathered  2^ 
bu.,  the  second  2|  bu.,  and  the  third  3f  bu.  How  many  bushels 
all  together  ? 

24.  If  wheat  sold  at  61|^  per  bushel  in  the  morning  and  the 
price  advanced  |^  during  the  day,  find  the  price  in  the  evening. 


Exercise  92 

fin 

id  the  sum  of : 

1. 

*,  i,  i- 

7. 

h  A,  h  ii- 

2. 

I  A.  f 

8. 

li3f,2A;4f 

3. 

t>  T7J>  T5' 

9. 

7A,  ^^,  m  ^' 

4. 

i  i  A- 

10. 

3t\,1A,5«,4||. 

5. 

h  h  H- 

11. 

2^,  14if ,  92f ,  15A- 

6. 

5>  TZi  ISJ  ^* 

12. 

19f  28H,  72|i,  59^. 

Exercise  93 

1.  Three  families  ordered  a  carload  of  coal.  The  first  got  8| 
T.,  the  second  7f  T.,  and  the  third  8|  T.  Find  the  amount  of 
coal  on  the  car. 

2.  A  farmer  raised  12S^^  bu.  of  wheat  from  one  field,  214|  bu. 
from  another,  and  from  a  third  field  156|  bu.  Find  his  total 
crop  of  wheat. 

3.  A  basket  weighing  ^^  lb.  contains  3J  lb.  coffee,  4f  *lb. 
butter,  and  5  lb.  6  oz.  cheese.    Find  the  total  weight. 


SUBTRACTION   OF  FRACTIONS  141 

4.  The  sides  of  a  field  are  24|  rd.,  42|  rd.,  mj\  rd.,  and  38| 
rd.     Find  its  perimeter. 

5.  What  is  the  combined  weight  of  three  men,  the  first  of 
whom  weighs  125 1  lb.,  the  second  147f  lb.,  and  the  third  175|  lb.  ? 

6.  A  grocer  has  three  barrels  of  oil;  the  first  contains  18| 
gal.,  the  second  24f  gal.,  and  the  third  16|  gal.  Find  how  much 
there  was  in  the  three  barrels. 

7.  Four  farms  join  each  other ;  the  first  contains  125f  A.,  the 
second  78|  A.,  the  third  96^  A.,  and  the  fourth  110|  A.  F'ind 
the  total  area. 

8.  I  paid  $125 J  for  a  horse  and  $57 J  more  than  that  for  a 
carriage.     Find  the  cost  of  both. 

9.  A  merchant  sold  four  pieces  of  cloth  containing  respec- 
tively 12i  yd.,  15f  yd.,  16J  yd.,  24|  yd.  How  many  yards  did  he 
sell  ?     What  did  he  receive  for  the  cloth  at  48  ^  a  yard  ? 

10.    From  a  piece  of  cloth  there  were  sold  3J  yd.,  7|  yd.,  8f 
yd.,  121  yd.     Find  the  total  value  of  the  cloth  at  72^  a  yard. 

SUBTRACTION   OF   FRACTIONS 

100.    (1)  Find  the  difference  between  |  hr.  and  -^^  hr. 
The  L.  C.  D.  =  30. 

5  8  ^  25  -  16  _  9    ^^    S^ 

6  15  30      ~30   ^^  10* 

.  *.  the  difference  =  ^-^  hr. 

Prove  this  result  by  expressing  f ,  ^\,  and  j^^  hr.  in  minutes,  and  taking 
the  difference  of  the  first  two. 

(2)  Find  the  value  of  |6||  -  |4JJ. 

6-4  =  2. 

The  L.  C.  D.  =  144. 

29      17  ^116 -51  _  65 
36     48  144      ~144* 

•  •.  the  difference  =  $2^^. 


142  ARITHMETIC 

(3)  Subtract  6^  da.  from  9f  da. 

8-6  =  2;  1^  =  ^. 

11       7  _  33 -14  ^19 
8       12  24  24* 

.*.  the  difference  =  2^1  da. 

On  reducing  ^^  and  f  to  24ths,  it  is  found  that  ^j  is  greater  than  |.    Hence 
we  break  9|  up  into  8  +  1^,  and  then  subtract  C  from  8  and  ^^  from  If  or  ^. 

Exercise  94 
Find  the  value  of : 

1.  ^A-^1%.  4.    f  gal. -i  gal. 

2.  3^  ft. -3-3^  ft.  5.    f^hr.-ifhr. 

3.  I  pk.  —  f  pk.  6.    ^J  da.  —  i^^  da. 

7.  What  is  the  direct  unit  of  measure  in  each  question?    Ex- 
press your  answer  in  two  ways. 

8.  Prove  results  by  reducing  each  fraction  to  the  next  lower 
denomination  and  then  subtracting. 

Exercise  95 
Find  the  difference  as  a  fraction  of  any  unit  of  measure : 

1.  |-|.  3.    f-|.  5.    H-f- 

2.  i-f.  4.    A-tV  6.    H-A- 

7.  Give  the  three  steps  required  in  subtracting  one  proper 
fraction  from  another. 

8.  In  Exercise  88,  examples  4  through  16,  subtract  the  smaller 
fraction  from  the  larger. 

9.  12-4.  14.    12-8H.  19.    m-SH- 

10.  5-^7^.  15.  8,^-5,^.  20.  4/^-lH. 

11.  8-^V  16.  9|^-4J|.  21.  46|-39f 

12.  7-4^V  17.  18^-12/^.  22.  9oJ-84f. 

13.  9-6if.  18.  4^-2J4. 


SUBTRACTION  OF  FRACTIONS 


143 


Exercise  06 

1.  A  dealer  bought  chickens  at  9^p  per  pound  and  sold  them 
for  12  J^  a  pound.  Find  his  gain  per  pound.  Find  his  gain  on 
selling  a  6-lb.  chicken. 

2.  A  dealer  bought  turkeys  at  lOJ^  per  pound  and  sold  them 
at  14^  per  pound.     Find  his  gain  on  a  12-lb.  turkey. 

3.  Dec.  9,  1899,  wheat  sold  in  Minneapolis  at  66f^  per 
bushel,  and  in  Toledo  at  681^  per  bushel.  Find  the  difference 
in  price. 

4.  On  the  same  date  wheat  sold  in  New  York  at  74 J  ^  per 
bushel,  and  in  Liverpool  at  83|^  per  bushel.  Find  the  difference 
in  price. 

5.  Dec.  9,  1899,  wheat  sold  in  Chicago  at  69|^  per  bushel,  and 
a  year  previous  to  that  date  at  G^^j^  per  bushel.  Find  the  increase 
in  price. 

6.  The  prices  of  wheat  per  bushel  Friday,  March  19, 1900,  and 
one  year  previous  to  that  date,  are  given  in  the  following  table. 
Find  the  difference  in  price  in  each  case : 


Friday 

Year  Ago 

Friday 

Year  Ago 

Chicago    .     .     . 
Minneapolis .     . 
New  York     .     . 
St.  Louis .     .     . 

$0.66| 
.631 
.72| 
.69^ 

f^0.70i 
.68i 
.751 

.741 

Duluth  .     .     . 
Toledo  .     .     . 
Liverpool  .     . 

$0.66 

.72f 
.83 

$0.69 

.73i 
.791 

7.  I  bought  wheat  at  63f  ^  per  bushel  and  sold  it  for  65^j^. 
Find  my  gain  on  8000  bu. 

8.  The  average  selling  price  of  wool  in  Montana  in  1899  was 
17^^  a  pound,  and  in  1898  was  16|j^.  The  farmers  of  that  state 
sold  7,000,000  lb.  in  1899  j  how  much  did  they  gain  on  account  of 
the  increase  in  price  ? 


144  ARITHMETIC 

Exercise  97 

1.   I  of  the  value  of  a  horse  =  ^  60. 
■J-  of  the  value  of  a  horse  =  ? 
|-  of  the  value  of  a  horse  =  ? 
.-.  the  horse  is  worth  f  ? 

Fill  out  the  blanks. 

By  what  do  you  divide  $60?  By  what  do  you  multiply  the 
result  ? 

2.  If  f  of  the  value  of  a  farm  is  $9000,  what  is  the  value  of  \ 
of  the  farm  ?     What  is  the  farm  worth  ? 

3.  A  person  sold  a  cow,  gaining  ^  of  the  cost  price.  If  he 
gained  $  12,  what  did  the  cow  cost  him  ? 

4.  A  man  lost  in  business  |  of  his  property.  His  loss  was 
$4500;  what  was  his  property  worth? 

5.  A  boy  lost  |  of  his  marbles,  and  then  had  60  left.  What 
fraction  of  his  marbles  did  he  have  left?  How  many  had  he 
at  first? 

6.  If  J  of  the  cost  of  a  gallon  of  wine  is  $3,  what  was  the 
cost? 

7.  A  merchant  sold  potatoes  at  75^  a  bushel,  gaining  ^  of  the 
cost.  The  selling  price  was  what  fraction  of  the  cost?  What 
was  the  cost  price  of  the  potatoes  per  bushel  ? 

8.  A  merchant  sold  cloth  at  90^  a  yard  and  gained  |  of  the 
cost.     Find  the  cost. 

9.  A  merchant  sold  cloth  for  80  ^  a  yard,  thereby  losing  \  of  the 
cost.     Find  the  cost. 

MULTIPLICATION  OF   FRACTIONS 

101.    (1)  Find  the  cost  of  12  yd.  of  cloth  at  $f  per  yard. 

We  are  required  to  find  the  quantity  measured  by  the  number  12  and  the 
measuring  unit  $  J. 

12  yd.  cost  V  X  $1  =  $9. 


MULTIPLICATIOK  OF  FRACTIONS  145 

(2)  Find  the  cost  of  f  yd.  at  $12  a  yard. 
The  cost  is  measured  by  the  number  f  and  the  unit  $  12. 

I  yd.  costs  f  of  $  12  =  $  10. 

(3)  Find  the  area  of  a  floor  12  ft.  long  and  9  ft.  9  in.  wide. 

The  area  is  measured  by  the  unit  9f  sq.  ft.,  which  is  the  area  of  1  strip, 
and  the  number  12. 

The  area  of  1  strip  =  9|  or  -\9-  sq.  ft. 

.-.  the  total  area  =  ^^^  x  ^^  sq.  ft.  =  117  sq.  ft. 

(4)  Reduce  |  ft.  to  inches. 

4 

^U.  =  -x^m.  =  ^  in.  =  10|  in. 
9^13  ^ 

3 

Exercise  98 
Find  the  cost  of : 

1.  10  yd.  at  $f  per  yard.  3.   9  yd.  at  1 2 J  a  yard. 

2.  12  yd.  at  f  If  per  yard.  4.   f  yd.  at  $  6  a  yard. 

5.    If  the  cost  price  is  measured  by  the  number  |  and  the  unit 
$  5,  find  the  cost. 

Find  the  cost  of : 

6.   A  4-lb.  chicken  at  9^^  a  pound. 

A  5-lb.  pail  of  butter  at  24^^  a  pound. 

A  13-lb.  turkey  at  111 ^  a  pound. 

9  lb.  cheese  at  12-|^  a  pound. 

A  bale  of  cotton  (1  bale  =  500  lb.)  at  9f  ^  a  pound. 

A  bale  of  cotton  at  9y\^  a  pound. 

233  bales  of  cotton  at  8|^  a  pound. 

A  10-lb.  ham  at  8|^  a  pound. 

8  lb.  fish  at  6|  ^  a  pound. 

12  doz.  eggs  at  12i/  a  dozen. 

Reduce  to  inches : 

7.    I  ft.;   I  ft.  8.    2|  ft.;  If  ft.  9.    Sf  ft.;   2|  ft. 


146  ARITHMETIC 

Reduce  to  yards : 

10.  8  rd.;  10  rd.;  12  rd.;  3  rd.;  9  rd.j  17  rd. 

Reduce  to  feet : 

11.  4rd.;  8rd.;  10  rd.;  3rd.;  7  rd.;  13  rd. 

12.  Find  the  cost  of  4  rd.  wire  fencing  at  2|^  a  foot. 

13.  Find  the  cost  of  the  wire  fencing  at  3|^  a  foot,  needed  to 
enclose  a  square  garden  2  rd.  on  a  side. 

Reduce  to  quarts : 
.  14.   3f  gal.;  2^  gal.;  5^^  gal.;  12J1  gal. 

15.  Find  the  cost  of  2 J  gal.  of  milk  at  6  ^  a  quart. 

Find  the  areas  of  the  floors  of  the  following  rooms : 

16.  Length  Width  Length  Width 
24  ft.         14  ft.  10  in.                   19  ft.        16  ft.  4  in. 
14  ft.         12  ft.  6  in.                     22  ft.         16  ft.  9  in. 

Find  the  area  of  the  walls  of  a  room  whose : 

17.  Perimeter  is  80  ft.,  height  9  ft.  6  in. 
18. .  Perimeter  is  63  ft,  height  10  ft.  8  in. 
19.   Perimeter  is  m  ft.,  height  8  ft.  9  in. 

102.    (1)  Find  the  cost  of  12f  yd.  of  cloth  at  $  If  a  yard. 

12f  yd.  cost  12|  X  $lf  =  ^^  X  $  V  =  $  V^  =  ^17H  =  $17.53. 

(2)  Find  the  area  of  the   four  walls  of  a  room  whose 
perimeter  is  62  ft.  8  in.,  and  height  8  ft.  9  in. 

The  perimeter  =  62  ft.  8  in.  =  62f  ft.  =  -4^  ft. 
The  height  =  8  ft.  9  in.  =  8f  ft.  =  »^  ft. 

47 
.-.  the  area  =  ^  x  ^sq.  ft.  =  648J  sq.  ft. 

(3)  Reduce  |  rd.  to  yards. 

4 
|rd.=|x5iyd.=?x:^  =  for4|yd. 


MULTIPLICATION  OF   FRACTIONS  147 


Exercise  99 

Find  the  value  of : 

1.    3fxflf 

1  X  1  lb. 

llxlA 

2ixf5i 

1  X  A  OZ. 

2|X4| 

^x$^ 

1  X  f  yd. 

3A  X  HI 

lix$2f 

fxjyd. 

4tV  X  !« 

3f  x$2i 

A  X  A  mi. 

5f  X  lA 

2.   Find  the  cost  of  a 

remnant  of  oilcloth  containing  5^  yd.  at 

6|)^  a  yard. 

Find  the  cost  of : 

3.   8^  lb.  cheese  at  9|  ^  a  pound. 
12^  lb.  turkey  at  12i  ^  a  pound. 
11^  lb.  turkey  at  10 J  ^  a  pound. 
20f  lb.  chickens  at  lOi  ^  a  pound. 
S^  lb.  veal  at  7|^  ^  a  pound. 
41  lb.  butter  at  221^  a  pound. 

4.  I  bought  cotton  at  8 J  ^  a  pound  and  sold  it  for  9f  /  a  pound. 
Find  my  gain  on  one  bale  (1  bale  =  500  lb.). 

5.  I  bought   cotton   at  8^^  a  pound  and  sold  it  for  9j^^  a 
pound.     Find  my  gain  on  600  bales. 

Eeduce  to  yards : 

6.  ^rd.;  l,^rd.;  f  rd.;  f  rd.;  |rd.;  2ird.;  5,^rd.;  S^rd. 

7.  Find  the  cost  of  9^^  rd.  of  wire  fencing  at  11^^  a  yard. 

8.  Find  the  cost  of  -^  rd.  of  wire  fencing  at  2|^  a  foot. 

Find  the  cost  of  the  following : 

9.  8^  yd.  at  $  31  a  yard ;  4f  yd.  at  ^  2f  a  yard ; 
20|  yd.  at  1 7i  a  yard ;  5i  yd.  at  ^  2|  a  yard. 


148  ARITHMETIC 

10.  21|  lb.  of  sugar  at  5^  ^  a  pound ; 
13J  lb.  of  sugar  at  5J-^  a  pound. 

11.  17|  yd.  of  cotton  at  11}^  a  yard. 

12.  151  doz.  of  eggs  at  14|  ^  a  dozen. 

13.  SJT.  of  hay  at  ^  Ufa  ton. 

14.  Find  the  area  of  a  floor  whose  dimensions  are :  (1)  lOJ  ft., 
9|  ft. ;  (2)  14^  ft.,  13^  ft. ;  (3)  15f  ft,  12|  ft. 

15.  Find  the  area  of  the  ceiling  of  a  room  whose  dimensions 
are :  (1)  12  ft.  6  in.,  10  ft.  8  in. ;  (2)  16  ft.  4  in.,  11  ft.  3  in. ; 
(3)  18  ft.  8  in.,  15  ft.  3  in. 

16.  At  40^  a  square  yard  find  the  cost  of  plastering  the  ceiling 
of  a  hall  16|  yd.  long  and  12f  yd.  wide. 

17.  Find  the  area  of  the  four  walls  of  a  room  whose  perimeter 
and  height  are  respectively :  (1)  52  ft.  6  in.,  9  ft.  4  in. ;  (2)  63  ft. 
4  in.,  10  ft.  6  in. 

18.  Draw  a  line  16  in.  long.  Mark  off  |  of  it.  The  remainder 
is  what  part  of  the  line  ?  Mark  off  |  of  the  remainder.  This  is 
what  part  of  the  whole  line  ?     |  of  f  =  ? 

19.  If  I  draw  a  line  and  mark  off  |  of  it,  and  then  mark  off  | 
of  the  remainder,  what  part  of  the  whole  line  will  I  mark  off  the 
second  time  ?  Illustrate  this  by  drawing  a  line  and  marking  it 
off.     2  of  3  =  ? 

20.  On  f  of  a  field  I  planted  potatoes  ;  on  |  of  the  remainder  I 
sowed  wheat.     What  part  of  the  field  did  I  sow  with  wheat  ? 

21.  I  withdrew  from  the  bank  |  of  my  deposit  and  then  f  of 
the  remainder.  What  pai-t  of  the  original  deposit  did  I  take  out 
the  second  time  ? 

22.  A  man  who  owns  ^^  of  a  ship  sells  J  of  his  share.  Wliat 
fraction  of  his  former  share  does  he  still  own  ?  What  fraction 
of  the  ship  ?  If  he  had  sold  J  of  his  share,  what  part  of  the  ship 
would  he  have  still  owned  ? 


CANCELLATION  149 

23.  A  grain  dealer  invested  f  of  his  money  in  wheat,  and  f  of 
the  remainder  in  oats.  What  part  of  his  money  did  he  invest  in 
oats  ?  If  he  invested  f  3000  in  oats,  how  much  did  he  have  at 
first  ? 

24.  The  owner  of  a  farm  valued  at  $  12,009  sells  f  of  it  to  one 
man,  and  ^  of  the  remainder  to  another.  What  part  of  the  farm 
does  he  sell  to  the  second  man,  and  what  should  he  get  for  it  ? 

25.  Four  brothers  enter  into  partnership ;  the  eldest  puts  in  ^ 
of  the  capital  and  the  others  the  remainder  in  equal  shares. 
W^hat  part  of  the  entire  capital  does  each  of  the  younger  brothers 
put  in  ?     If  they  each  put  in  $  2000,  what  is  the  entire  capital  ? 

26.  If  I  own  f  of  I  of  a  business,  what  part  do  I  own  ?  If  I 
sell  ^  of  my  share,  what  part  of  the  business  do  I  still  own  ? 

27.  A  man  left  his  farm  to  be  divided  among  his  three  sons ; 
the  oldest  got  80  A.,  the  second  ^  of  the  farm,  and  the  youngest 
f  as  much  as  the  other  two.  Prove  that  the  farm  contained 
210  A. 

28.  If  the  loss  is  measured  by  the  number  |  and  the  unit  $  17|, 
what  is  the  loss  ? 

If  the  gain  is  measured  by  the  number  ^  and  the  unit  $  11|, 
what  is  the  gain  ? 

CANCELLATION 

103.    (1)  Find  the  product  of  if  x  6f  x  7^. 

Reducing  to  improper  fractions  and  cancelling,  the  product 
8       19 

(2)  Find  the  volume  of  a  solid  whose  dimensions  are  2| 
in.,  2|  in.,  and  4^  in. 

The  volume  =  2f  x  2f  x  4|  cu.  in. 
=  ^/  X  f  X  f  cu.  in. 
=  ^^  or  31^  cu.  in. 

How  do  you  find  the  volume  of  a  rectangular  solid  ? 


150  ARITHMETIC 

Exercise  100 

In  the  following  exercise,  indicate  the  operations  as  in  the  pre- 
ceding paragraph,  and  cancel  where  possible : 

Find  the  volume  of  the  solids  whose  dimensions  are : 

1.  2  in.,  4  in.,  61  in.  4.   ^  ft.,  2|  ft.,  4|  ft. 

2.  Hin.,  2iin.,  2Jin.  5.   |  ft.,  f  ft.,  ^^  ft. 

3.  2j\in.,2^\m.,4.iin.  6.   /^  yd.,  2^  yd.,  8^  yd. 

Find  the  product  of : 

7.   VxfX-A-.  10.  Hx2Axl3f. 

8-  fVxfxIf.  11.  ^x2|x3||. 

9-  «xixf|.  12.  16Jx4^x2^. 

13.  Find  the  weight  of  a  solid  4^  in.  long,  IJ  in.  wide,  |  in. 
thick,  if  1  cu.  in.  weighs  ^  oz. 

14.  Find  the  weight  of  a  piece  of  lumber  10  ft.  long,  |  ft.  wide, 
i  ft.  thick,  if  1  cu.  ft.  weighs  21f  lb. 

15.  Find  the  weight  of  f  of  a  piece  of  lumber  16  ft.  long,  |  ft. 
wide,  i  ft.  thick,  if  1  cu.  ft.  weighs  ISy^  lb. 

16.  What  is  the  weight  of  a  piece  of  floor  oilcloth  1\  yd.  long, 
1^  yd.  wide,  if  1  sq.  yd.  weighs  31  lb.  ? 

17.  AVhat  is  the  weight  of  one  dozen  floor  oilcloths,  each  2J 
yards  square,  weighing  3J  lb.  to  the  square  yard  ? 

18.  If  an  acre  of  land  produces  42  bu.  of  potatoes,  how  much 
will  J  of  -f^  of  an  acre  produce  ?  What  is  their  value  at  40  ^  a 
bushel  ? 

19.  If  an  acre  of  land  produces  43^  bu.  of  potatoes,  what  is  the 
value  of  the  potatoes  grown  on  |  of  |  A.  at  35  ^  a  bushel  ? 

20.  If  a  certain  number  of  men  can  do  a  piece  of  work  in  24 
da.,  how  long  will  it  take  them  to  do  a  similar  piece  of  work  f  as 
hard  and  J  as  great  ?  How  long  if  |  as  hard  and  IJ  times  as 
great  ? 


DIVISION  OF  FRACTIONS  151 

21.  If  a  certain  number  of  men  can  do  a  piece  of  work  in  18 
da.,  how  many  days  will  they  take  to  do  a  similar  piece  1^  times 
as  hard  and  1^  times  greater,  if  they  work  -f*^  as  long  each  day  ? 

DIVISION  OF  FRACTIONS 


104.  The  quantity  AB,  in  the  above  drawing,  is  repre- 
sented by  I  of  itself.  We  think  of  the  quantity  as  made  up 
of  5  units,  and  the  unit  as  equal  to  |-  of  the  quantity.  Hence 
we  may  think  of  any  quantity  as  equal  to  5  x  ^  (z.e.  five 
times  one-fifth)  of  itself. 

What,  then,  is  the  meaning  of  the  terms  5  and  ^  considered 
separately  ?  5  shows  the  ratio  of  the  quantity  to  the  unit  of 
measure^  and  ^  shows  the  ratio  of  the  unit  of  measure  to  the 
quantity.  Numbers  thus  mutually  related  are  said  to  be 
reciprocal. 


Again,  let  the  line  AB,  which  is  divided  into  5  equal  parts,  represent  the 
primary  unit,  $1.  Then  AC  represents  the  quantity  denoted  by  $f.  Hence 
it  is  evident  from  the  diagram  that  f  is  the  ratio  of  AG  to  AB,  i.e.  of  tlie 
quantity  denoted  by  $  f  to  the  primary  unit,  $  1 ;  also,  that  its  reciprocal  f 
is  the  ratio  of  AB  to  AC,  i.e.  of  the  primary  unit,  $1,  to  the  quantity  de- 
noted by  $f.  Similarly,  the  ratio  of  $1.  (which  contains  2  units)  to  the 
quantity  $f  (which  contains  3  units)  is  equal  to  |,  i.e.  to  the  reciprocal  of  |. 


Exercise  101 

1.  Draw  a  line,  AB,  divided  into  6  equal  parts.  The  line  AB 
is  represented  by  how  many  units  ?  by  what  fraction  of  itself  ? 
What  is  the  ratio  of  AB  to  the  unit  ?  of  the  unit  to  AB  ?  6  and 
i  are  called  reciprocals  of  each  other. 

2.  What  is  the  reciprocal  of5?3?4?6?8?9? 

3.  What  is  the  reciprocal  ofi?  i?  i?  i?  \?  j\? 


152  ARITHMETIC 

4.  What  is  the  reciprocal  of  3?  J?  i?  7?  -^^?  ^7 

5.  What  is  the  ratio  of  \  ft.  to  1  ft.  ?  Of  1  ft.  to  \  ft.  ?  Show 
by  a  drawing. 

6.  What  is  the  ratio  of  \  ft.  to  1  ft.  ?  Of  1  ft.  to  \  ft.  ?  Of 
^Jtofl?     Of^lto^i?     Of  1  yd.  to  I  yd.  ?    Of  ^  yd.  to  1  yd.  ? 

7.  What  is  the  ratio  of  1  qt.  to  \  qt.  ?  Of  ^  gal.  to  1  gal.  ?  Of 
\  mi.  to  1  mi.  ?  Of  1  T.  to  ^  T.?  Of  1  da.  to  yV  <ia.?  Of  ^ 
min.  tol  min.?     Ofjij-tol^     Oflto^j? 

8.  Multiply  I  by  the  reciprocal  of  6. 

9.  Multiply  I  by  the  reciprocal  of  each  of  the  following  num- 
bers :  3,  6,  9,  15,  2,  18,  5,  11. 

10.  Multiply  IJ  by  the  reciprocal  of  J. 

11.  Multiply  2|  by  the  reciprocal  of  each  of  the  following  num- 
bers: i,  i,  i,  iVj  i,  tV -f ,  yV 

12.  Multiply  I  by  the  reciprocal  of  each  of  these  numbers :  4, 
4,  A,  6,  i,  8, 12,  i. 

13.  Draw  a  line,  ^JB,  1  ft.  long.  Mark  off  AC  equal  to  |  ft. 
What  is  the  ratio  of  AC  to  AB  ?  What  is  the  ratio  of  AB  to  AC? 
What  is  the  ratio  of  f  ft.  to  1  ft.  ?  Of  1  ft.  to  f  f t.  ?  |  and  J  are 
called  reciprocals  of  each  other. 

14.  What  is  the  ratio  of  |  ft.  to  1  ft.?  Of  1  ft.  to  |  ft.? 
Show  by  a  drawing. 

15.  What  is  the  ratio  of  I  yd.  to  1  yd.  ?  Of  J  lb.  to  1  lb.  ?  Of 
I  da.  to  1  da.  ?     Of  f  wk.  to  1  wk.  ? 

16.  What  is  the  ratio  of  1  mi.  to  |  mi.  ?  Of  1  T.  to  ^  T.  ?  Of 
1  A.  to  I  A.?     Of  1  A.  to  f  A.? 

17.  What  is  the  ratio  of  f  lb.  to  1  lb.  ?  Of  1  oz.  to  |  oz.  ?  Of 
1  gal.  to  ^  gal.  ?  Of  I  qt.  to  1  qt.  ?  Of  1  doz.  to  ^  doz.  ?  Of  $  1 
to  If? 

18.  What  is  the  reciprocal  of|?f?|?f?J?}?f?J/? 
If?  2i?  IJ?  ^? 


DIVISION  OF  FRACTIONS  153 

19.  Multiply  f  by  the  reciprocal  of  |.  Multiply  f  by  the  recip- 
rocal of  \^. 

20.  Multiply  f  by  the  reciprocal  of  each  of  the  following  frac- 
tions :  "2,  8,  -Q,  ^j  I",  '^2,  -g->  ^4?  1 8"?  e"" 

Exercise  102 

1.  Make  drawings  to  show  that  (1)  1  ft.  --  J  ft.  =  4.  (2)  1  ft. 
-Jft.  =  6. 

2.  1  ft.  -=-  i  ft.  =  ?        1  A.  -  i  A.  =  ?        1  da.  -  j\  da.  =  ? 
lyd.-iyd.  =  ?       1T.-3VT.  =  ?         1  yr.  -  J^  yr.  =  ? 

1  lb.  -  ^  lb.  =  ?      1  bu.  -  1  bu.  =  ? 

3.  How  many  meals  will  1  bu.  of  oats  last  a  horse,  if  he  has 
|-  bu.  a  meal  ? 

4.  Into  how  many  lots,  each  ^  A.  can  you  divide  1  A.  ? 

5.  Make  drawings  to  show  that 

(1)  3  ft.  -  J  ft.  =  12  (2)  21  ft.  -f-  1-  ft.  =  10 

(3)  2|  ft.  --  i  ft.  =  8  (4)  1|  ft.  H-  i  ft.  =  10 

6.  Multiply  3  by  the  reciprocal  of  J ;  2i  by  the  reciprocal  of  ^^ 
2|  by  the  reciprocal  of  i ;  1|  by  the  reciprocal  of  ^.  What  is  the 
result  in  each  case  ? 

By  what  do  you  multiply  2J  to  find  the  value  of  2|  -j-  ^? 

7.  2  ft.  -f- 1-  ft.  =  ?  3|  -^  i  =  ? 
3ift.-ift.  =  ?  7i^i  =  ? 

4Jyd.^iyd.  =  ?  -    3i^i  =  ? 

5ihr.-ihr.  =  ?  2i-i  =  ? 

2|  da. -^  3-V  da.  =  ?  3J  ^  J  =  ? 

8.  Show  by  a  drawing  that  2|-  -j-  ^  =  6J. 

9.  A  merchant  put  up  7^  lb.  tea  in  packages,  each  containing 
J  lb.     Find  the  number  of  packages. 


154  ARITHMETIC 

10.  A  piece  of  ribbon  3-J-  yd.  long  is  cut  into  parts  each  ^  yd. 
long.     How  many  parts  ? 

11.  Draw  a  line  4  ft.  long.  Divide  it  into  parts  each  J  ft.  long. 
How  many  ?  How  many  of  these  12  parts  make  |  ft.  ?  How 
many  parts,  each  |  ft.  long,  can  you  count  in  these  12  parts  ? 

4  ft.  -- 1  ft.  =  ?  6  ft.  -- 1  ft.  =  ?  " 

12.  In  the  first  part  of  example  11,  by  what  did  you  multiply 
4  to  get  12  ?     By  what  did  you  divide  12  to  get  6  ? 

13.  How  can  you  divide  a  number  like  4  by  a  fraction? 
{To  divide  a  number  by  a  fraction  multiply  the  number  by  the 
reciprocal  of  the  fraction.) 

14.  Find  the  quotient  in  each  of  the  following  examples : 
9  ft.  -- 1  ft.  ^  6  --    $  I  5  --    J 

8  yd. -I  yd.  ^12-^   I  7 

12  mi.  --  f  mi.  $  10  h-  $   |  9 

16  lb.  -^  I  lb.  $  10  -  $  1 J  21  -  3i 

12T.-5-fT.  ^15 --^2^  16 

15.  How  many  strips  of  carpet  f  yd.  wide  are  needed  to  carpet 
a  room  6  yd.  wide  ?  If  the  room  is  8  yd.  long  how  many  yards 
of  carpet  are  needed  for  it  ? 

16.  What  part  of  a  yard  is  32  in.  ?  How  many  strips  of  hemp 
carpet  32  in.  wide  are  needed  to  carpet  a  hall  16  yd.  wide  ?  If 
the  hall  is  20  yd.  long,  how  many  yards  of  carpet  are  needed  ? 

17.  As  in  examples  5  and  8,  make  a  drawing  to  show  that 
2i  ft.  ^  f  ft.  =  3. 

18.  How  can  you  find  the  value  of  2\  -i-  J,  without  actually 
dividing  one  quantity  by  another  ? 

19.  To  divide  one  fraction  by  another ^  multiply  the  first  fraction 

by  the  reciprocal  of  the  second. 


DIVISION  OF   FRACTIONS  155 

20.    Find  the  value  of : 


1 

^i 

f 

•^f 

f 

-^1 

A 

^¥ 

A 

^1 

21.    . 

A.tl2i 

414 

m 
n 

45| 


4| 

-5i 

74 

^5i 

5| 

■^H 

23f 

■^H 

6A 
4i 


2i 
14| 


a  pound,  how  many  pounds  of  chicken  will  cost 
50^? 

22.  Find  the  number  of  square  yards  in  a  piece  of  floor  oil- 
cloth, weighing  7^  lb.  at  2^  lb.  per  square  yard.  At  li  lb.  per 
square  yard. 

23.  A  double  roll  of  wall  paper  weighs  IJ  lb.  Find  how  many 
double  rolls  weigh  8|  lb. 

24.  How  many  double  rolls  of  wall  paper  weigh  10-J-  lb.  ? 
Find  their  cost  at  25  ^  a  double  roll. 

25.  Find  the  cost  of  a  parcel  of  wall  paper,  containing  15 J  lb., 

at  20^  a  double  roll. 

26.  Find  the  quotients  : 

331  ^  10  330f  -r-  21  352^  -j-  9^ 

37|  -  9  2331  -  81  230|  -  11^ 

105.    (1)  How  do  you  reduce  feet  to  yards  ? 
Reduce  2^  ft.  to  a  fraction  of  1  yd. 

21ft.  =2|-3or|  xiyd.  =fyd. 

(2)  Find  the  number  of  strips  required  to  carpet  a  room 
12  ft.  wide  with  carpet  21  ft.  wide. 

The  number  of  strips  =  12  ft.  -f-  2^  ft.  =  i^  x  ^  =  4|,  i.e.  5. 

.  •.  5  strips  are  required. 


156  ARITHMETIC 

106.    (1)  2  yd.  of  cloth  cost  72^,  what  will  1  yd.  cost? 

(2)  3  yd.  of  cloth  cost  75^,  what  will  1  yd.  cost  ? 

(3)  I  yd.  of  cloth  cost  48^,  what  will  1  yd.  cost  ? 

In  example  (1)  what  is  the  divisor  of  72/?  ?    In  example  (2)  what  is  3  ? 
In  example  (3)  what  is  |  ? 

I  yd.  costs  48  ^. 

1  yd.  costs  48  j?  H-  I  =  48  ^  X  I  =  64  f 

In  this  example  the  whole  measured  quantity  is  48  ^,  and  the  number  is  |. 
We  find  the  measuring  unit  by  dividing  48  f  by  |. 

Exercise  103 
Find  the  number  of  pounds  of : 

1.  Veal  at  7|^  a  pound  that  cost  45^. 

2.  Chicken  at  8J ^  a  pound  that  cost  h2\<^, 

3.  Cheese  at  12f  ^  a  pound  that  cost  $1.27f 

4.  Turkey  at  12^^  a  pound  that  cost  %  1.56J. 

Find  the  number  of : 

5.  Barrels  of  flour  at  $  5f  a  barrel  that  cost  $103^. 

6.  Eods  of  barbed  wire  fencing  in  a  roll  weighing  25^  lb.,  if 
1  rd.  weighs  ly^^  lb. 

7.  Yards  of  carpet,  weighing  24  oz.  to  1  yd.,  in  a  roll  weighing 
Hi  lb. 

8.  f  of  the  time  I  took  to  do  a  piece  of  work  is  3%  of  an  hour. 
How  long  did  I  take  to  do  it  ?     {-^-^  hr.  -f- 1.) 

9.  I  yd.  of  cloth  cost  %  f ,  what  will  1  yd.  cost  ? 

10.  }  yd.  of  cloth  cost  %  f ,  what  will  1  yd.  cost  ? 

11.  f  yd.  of  carpet  weighs  f  lb.,  find  the  weight  of  1  yd. 
How  many  ounces  ? 


DIVISION  OF  FRACTIONS  157 

12.  •  If  f  yd.  of  carpet  weighs  1-J-  lb.,  how  many  ounces  will  1  yd. 
weigh  ? 

13.  If  21  doz.  silver  forks  weigh  3^  lb.,  how  many  pounds  will 
1  doz.  weigh  ?     How  many  ounces  ? 

14.  Eeduce  to  the  fraction  of  a  foot :  1^  in.,  2J  in.,  4^  in.,  2|  in. 

15.  Eeduce  to  the  fraction  of  a  yard :  1^  ft.,  2|  ft.,  1^  ft.,  21  ft. 

16.  What  is  the  cost  of  a  piece  of  carpet  2^  ft.  long  at  80^  a 
yard? 

17.  Eeduce  to  fractions  of  one  dollar:  $.12i  $.371  $.621 
$.871  $.16|,  $.061  $.06|. 

18.  At  one  dollar  a  day,  how  long  will  it  take  a  boy  to  earn 
$.371?     $1,621? 

19.  Find  the  lengths  of  the  following  rooms  : 

Area  300  sq.  ft.,  width  15  ft. 
Area  466|  sq.  ft.,  width  20  ft. 
Area  2421  sq.  ft.,  width  15  ft. 
Area  681^  sq.  ft.,  width  25  ft. 
Area  8771  gq.  ft.,  width  27  ft. 

20.  Eeduce  to  the  fraction  of  a  rod :  2|  yd..  If  yd.,  41  yd., 
6|  yd.,  4^\  yd. 

21.  Find  and  prove  by  actual  measurement  the  following: 
2i  ft. -^1  ft. ;  11  ft. -1  ft. ;  2i  ft. -1  ft. ;  If  ft. -J  ft. ;  4^  ft.  -^^\  ft. 

22.  Find  the  number  of  strips  of  carpet  required  to  carpet  each 
of  the  following  rooms : 

Width,  22  ft.  6  in. ;  width  of  carpet,  27  in.     (22^  ft.^2i  ft.) 

Width,  15  ft.  9  in. ;  width  of  carpet,  27  in. 

Width,  10  ft.  8  in. ;  width  of  carpet,  32  in. 

Width,  14  ft.  8  in. ;  width  of  carpet,  32  in. 

23.  How  many  yards  of  carpet  are  needed  to  carpet  the  last 
room  in  the  preceding  example,  if  its  length  is  5  yd.  ?  6  yd.  ? 
21  ft.  ?    24  ft.  ? 


158  ARITHMETIC 

24.  27  in.  is  what  part  of  a  yard  ?  How  many  yards  of  carpet 
are  needed  to  carpet  a  room  8  yd.  long  and  6|  yd.  wide  with  carpet 
27  in.  wide,  the  carpet  running  lengthwise  ? 

Make  a  drawing  to  illustrate  this.     (Scale  1  in.  to  1  yd.) 

25.  How  many  yards  of  hemp  carpet,  32  in.  wide,  are  needed 
to  carpet  a  room  12  yd.  long  and  10  yd.  2  ft.  wide,  the  carpet  run- 
ning lengthwise  ?     Find  the  cost  of  this  carpet  at  24^  a  yard. 

26.  Find  the  length  of  each  of  the  following  rooms : 

Area  of  floor,  126|    sq.  yd. ;  width,  8  yd.  2  ft. 

Area  of  floor,  133 J |  sq.  yd. ;  width,  9|  yd. 

Area  of  floor,  334  L|  sq.  ft. ;  width,  15  ft.  9  in. 

Area  of  floor,  253^    sq.  ft. ;  width,  10  ft.  8  in. 

27.  Find  the  perimeter  of  each  of  the  following  rooms : 

Area  of  the  Walls  Height 

533J  sq.  ft.  8  ft.  4  in. 

704|  sq.  ft.  9  ft.  4  in. 

877^  sq.  ft.  10  ft.  9  in. 

65i  sq.  yd.  2  yd.  2  ft. 

107.   What  is  the  ratio  of  4  da.  to  6  da.  ? 

The  ratio  of  4  da.  to  6  da.  =  4  ^  6  =  |  =  |. 

What  is  the  ratio  of  2|  da.  to  3J  da.  ? 

The  ratio  of  2|  da.  to  3|  da.  =  2^  h-  3^  =  f  ^  -»^  =|  x  ^^  =  f.  What  is 
the  ratio  of  3^  da.  to  2^  da.  ?         - 

Exercise  104 
1.   What  is  the  ratio  of : 
1^  to  4  3|  to  2J 

22  to  6|  3J  to  3J 

6J  to  2f  I  to  I 

2J  to  3}  I  to  f 


Iff 

7H  to  17| 

AtorV 

18f  to  13J 

e^  to  7i 

IGi  to  23i 

17i  to  3| 

52i  to  18§ 

DIVISION  OF  FRACTIONS  159 

2.  What  is  the  ratio  of  2|  to  4|  ?  If  4f  T.  of  coal  cost  ^  27, 
what  part  of  $  27  will  2|  T.  cost  ?     How  much  ? 

3.  What  is  the  ratio  of  f  to  3^  ?     Of  3^  to  f  ? 

If  a  boy  rides  7  mi.  in  |  hr.,  how  far  can  he  ride  in  3J  hr.  at 
the  same  rate  ? 

4.  A  piece  of  matting,  weighing  2^-  lb.  to  the  square  yard, 
weighs  120  lb.  Find  the  weight  of  a  piece  of  the  same  size, 
weighing  3J  lb.  to  the  square  yard. 

5.  I  paid  $  20.80  for  silver  spoons,  at  $  6|  a  dozen.  Find  the 
cost  of  the  same  number  at  $  7^  a  dozen. 

6.  If  6f  T.  of  hay  cost  $42,  what  part  of  $42  will  l^f  T. 
cost?     How  much  will  l^f  T.  cost? 

7.  If  $  ^\  is  represented  by  unity,  what  number  will  represent 

$11? 

8.  Find  the  value  of  (1)  5|  h-  3f ;  (2)  7J  ^  4 Jj ;  (3)  2fj  -  5^ ; 
(4)111-2^. 


108.   Simplify   .  . 


4f 


5i 


Or  thus,  multiplying  the  numerator  and  denominator  by  12,  the  L.  C.  M. 
of  3  and  4,  we  have 

4|_66_8 
si  ~  63  ~  9' 


Exercise  104  (a) 

Simplify,  using  either  method : 


'•!• 

3  2i 

6     "* 
30i 

'k 

4     ^. 
3| 

7. 


8. 


6i 

m 

H 

25i" 


160  ARITHMETIC 

109.  (1)  A  owns  a  farm  containing  81|  A.,  B  owns  96^^  A., 
and  C  64 j^  A.     How  many  acres  do  they  own  all  together  ? 

Here  we  are  required  to  find  the  whole  quantity  measured  by  the  parts, 
.-.  the  sum  =  241  A.  +  h\\  A.  =  242^^  A. 

(2)  A  sum  of  money  is  divided  among  4  persons.  The 
first  receives  ^,  the  second  -|,  the  third  ^,  and  the  fourth  the 
remainder.  It  is  found  that  the  first  received  ^700  more 
than  the  fourth.     Find  the  sum  received  by  each. 

Consider  the  sum  of  money  as  made  up  of  60  units. 

The  first  receives  |  of  60  or  20  units;  the  second  15,  and  the  tliird  12 
units. 

The  first  three  receive  20  +  15  +  12  or  47  units. 

The  fourth  receives  60  —  47  or  13  units. 

The  first  receives  20  —  13  or  7  units  more  than  the  fourth. 

7  units  =  $  700. 

1  unit   =  $  100. 

.-.  the  first  receives  20  units  or  $2000,  the  second  $  1600,  the  third  3 1200, 
and  the  fourth  $  1300. 

Exercise  105 

1.  What  is  the  combined  weight  of  three  sheep,  the  first  of 
which  weighs  133^  lb.,  the  second  127f  lb.,  and  the  third  135}  lb.  ? 

2.  A  grocer  had  three  barrels  of  sugar ;  the  first  contained  28| 
lb.,  the  second  44|,  and  the  third  16f.  Find  how  much  there 
was  in  the  three  barrels. 

3.  If  three  crocks  contain  respectively  8 J,  12f,  and  14ff  lb.  of 
butter,  how  much  do  the  first  two  contain  more  than  the  tliird  ? 

4.  Four  farms  join  each  other ;  the  first  contains  125|-  A.,  the 
second  85|  A.,  the  third  156J  A.,  and  the  fourth  120f  A.  Find 
the  total  area. 


DIVISION  OF  FRACTIONS  161 

5.  A  lady  had  three  dresses  made.  The  first  contained  12i 
yd.,  the  second  14|  yd.,  and  the  third  16J  yd.  Find  the  average 
number  of  yards  in  a  dress. 

6.  A  merchant  sold  to  four  customers,  respectively,  15^  yd., 
14|  yd.,  16|  yd.,  and  18|  yd.  from  the  same  piece  of  cloth.  Find 
the  selling  price  at  72  ^  a  yard. 

7.  A  person  paid  $  165 J  for  a  horse,  and  $  23|-  more  than  that 
for  a  carriage,  and  shortly  after  sold  them  at  a  loss  of  $46f. 
What  was  the  selling  price? 

8.  How  much  greater  than  unity  is  If  ?  What  fraction  sub- 
tracted from  the  sum  of  f  and  ^  will  leave  unity  for  remainder  ? 

9.  If  f  of  a  school  term  exceed  ^  of  it  by  13^  da.,  how  many 
days  are  there  in  the  whole  term  ? 

10.  I  am  the  owner  of  ^  of  a  ship  worth  $  30,000,  and  sell  i  of 
the  ship.  What  part  of  it  will  then  belong  to  me,  and  what  will 
it  be  worth  ? 

11.  Add  together  the  greatest  and  least  of  the  fractions  f,  |,  -J, 
ij,  and  subtract  their  sum  from  the  sum  of  the  other  two  frac- 
tions. 

12.  From  a  piece  of  cloth  containing  45  yd.  there  were  sold  12f 
yd.,  13J  yd.,  and  15f  yd.  The  remainder  was  put  on  the  remnant 
table  and  sold  for  $  1.65.  Find  the  number  of  yards  it  contained 
and  what  it  sold  for  per  yard. 

13.  An  estate  worth  $  10,000  is  left  to  A,  B,  and  C ;  f  to  A,  | 
to  B,  and  the  remainder  to  C.     Find  C's  portion  and  its  value. 

14.  I  paid  15^  for  cheese  at  20  ^  a  pound.  How  many  ounces 
did  I  buy  ? 

15.  If  during  the  day  I  pay  out  |-,  then  ^  next  -^j,  and  lastly 
Y^g-  of  the  money  I  had  in  the  morning,  what  fraction  of  it  have  I 
left  ?    If  the  sum  left  amounts  to  $  1.54,  what  sum  had  I  at  first  ? 

16.  At  a  public  meeting  f  of  those  present  were  voters.  If 
there  were  135  voters  present,  how  many  attended  the  meeting? 


162  ARITHMETIC 

17.  At  an  entertainment  ^  of  the  seats  were  reserved  and  sold 
at  50^  each.  The  remamder  brought  ^40  at  25^  each.  Find 
the  number  who  attended  and  the  total  amount  received. 

18.  In  a  certain  subscription  list,  ^  of  the  number  of  subscrip- 
tions are  for  $  5  each,  ^  are  for  $  4  each,  ^  are  for  $  2  each,  ^ 
are  for  $  1  each,  and  the  remaining  subscriptions,  amounting  to 
$10.50,  are  for  50^  each.  Find  the  whole  number  of  subscribers, 
and  the  total  amount  of  their  subscriptions., 

19.  A  man  lost  \  of  his  property  in  speculation ;  he  afterwards 
purchased  a  partnership  in  business  for  $  16,000,  and  had  still 
$  6000  left.     What  was  he  worth  at  first  ? 

20.  A  house  and  lot  cost  $  3600 ;  the  value  of  the  lot  is  ^  that 
of  the  house.     Find  the  value  of  each. 

21.  What  must  be  the  length  of  a  plot  of  ground,  if  the  breadth 
is  15f  ft.,  that  its  area  may  contain  35  sq.  yd.  ? 

22.  A  has  $  6  more  than  ^  of  the  whole  of  a  sum  of  money ; 
B  has  $  8  more  than  ^  of  the  whole ;  and  C  has  $  12  more  than  J 
of  the  whole.     Find  the  sum  divided. 

110.    (1)  A  owns  f  of  a  ship  and  B  the  remainder,  and  | 

of  the  difference  between  their  shares  is  il500.     What  is 

the  vessel  worth  ? 

B's  share  =  f  of  the  ship. 

The  difference  =  ^  of  the  ship. 

I  of  the  difference  =  |  of  ^  or  ^  of  the  ship. 

^  of  the  ship  =  $1500. 

.-.  the  value  of  the  ship  =  $  1500  x  Y  =  $  10,000. 

(2)  A  man  lost  |  of  the  value  of  his  horse  by  selling  it  for 

f  60.    For  what  should  he  have  sold  it  to  gain  |  of  its  value  ? 

Let  the  value  of  the  horse  =  5  units  of  money. 

Then  the  loss  on  selling  =  2  units  of  money. 

The  first  selling  price  =  3  units  of  money. 

The  second  selling  price  =  7  units  of  money. 

.'.  the  second  selling  price  =  |  of  $  60  =  $  140. 


DIVISION  OF  FRACTIONS  163 

(3)  A  person  who  has  |  of  a  mine  sells  |  of  his  share  for 
6000.     What  is  the  value  of  the  wliole  mine  ? 


.-.  the  value  of  the  mine  =  $  6000  x  -\^  =  $  20,000. 

Exercise  106 

1.  A  grocer  buys  tea  at  64^  a  pound,  and  sells  it  so  as  to  gain 
-^  of  the  cost  price.  Find  what  he  receives  on  selling  a  chest  of 
tea  containing  88  lb. 

2.  If  I  own  f  of  f  of  I  of  a  ship  worth  |  20,000,  and  sell  J  of 
the  ship,  what  will  the  part  I  have  left  be  worth  ? 

3.  Find  the  number  of  square  feet  in  a  garden  9^  yd.  long 
and  5^  yd.  wide. 

4.  Find  the  cost  of  sodding  a  piece  of  ground  8^  yd.  long  and 
6^  yd.  wide  with  sod  costing  li^  per  square  foot. 

5.  A  city  lot  831  ft.  long  and  66^;  ft.  wide  sold  for  $  f  a 
square  foot.     Find  what  the  lot  sold  for. 

6.  A  man  has  f  4000  in  the  bank.  He  drew  out  ^  of  it,  and 
then  ^  of  the  remainder,  and  afterwards  deposited  ^  of  what  he 
had  drawn  out.     How  much  had  he  then  in  the  bank  ? 

7.  A  man  divided  a  farm  among  three  sons;  to  the  first  he 
gave  80  A.,  to  the  second  |  of  the  whole,  and  to  the  third  f  as 
much  as  to  both  the  others.  How  many  acres  did  the  farm 
contain  ? 

8.  Divide  $  65.80  between  two  persons,  so  that  one  shall  re- 
ceive one-third  as  much  again  as  the  other. 

9.  Five  brothers  join  in  paying  a  sum  of  money ;  the  eldest 
pays  -f  of  it,  and  the  others  pay  the  remainder  in  equal  shares. 
It  is  found  that  the  eldest  brother  pays  $  300  more  than  a  younger 
brother's  share.     Find  the  sum  of  money. 

10.  Show  by  measuring  the  diameter  and  the  circumference 
of  a  circle  (as  a  plate,  etc.)  that  the  circumference  is  about  3^ 
times  the  diameter. 


164  ARITHMETIC 

11.  What  is  the  circumference  of  each  of  the  following  circles, 
the  diameter  being : 

Tin.?  17^  in.?  12^  in.?  3.5  in.? 

21  in.  ?  15f  in.  ?  J  ft.  ?  1.26  in.  ? 

lO^in.?  Sljin.?  |ft.?  6.51  in.? 

12.  How  can  you  find  the  circumference  of  a  circle  when  the 
diameter  is  given  ?  How  would  you  find  the  diameter  if  the  cir- 
cumference were  given  ?. 

13.  What  is  the  diameter  of  each  of  the  following  circles,  the 
circumference  being : 

44  in.?  S^in.?  ly^ft.?  8.8ft.? 

66  in.?  3|in.?  15f  ft. ?  13.2  ft.? 

99  in.?  8|in.?  3|ft.?  17.6  ft.? 

14.  Show  clearly  that  both  terms  of  a  fraction  can  be  multi- 
plied by  the  same  number,  without  changing  the  value  of  the 
fraction. 

15.  A  man  commenced  business  with  a  capital  of  $  8000;  the 
first  year  he  gained  |  of  his  capital,  and  added  his  gain  to  his 
capital ;  the  second  year  he  gained  J,  adding  his  gain  as  before ; 
the  third  year  he  lost  ^  of  his  last  capital.  How  much  did  he 
make  in  the  three  years  ? 

16.  If  I  of  f  of  an  acre  produces  16  bu.  of  potatoes,  how  many 
bushels  will  an  acre  produce  ? 

17.  A  paid  $  60  per  acre  for  his  farm,  which  was  |  as  much  as 
B  paid  per  acre  for  his  farm  of  150  A.  Find  the  entire  cost  of 
B's  farm. 

18.  A  piece  of  cloth,  when  measured  with  a  yard  measure 
which  is  J  in.  too  short,  appears  to  be  4  yd.  long.  How  much 
shorter  is  it  than  4  yd.  ?     How  long  is  it  ? 

19.  A  piece  of  cloth,  when  measured  with  a  yard  measure 
which  is  I  of  an  inch  too  short,  appears  to  be  10^-  yd.  long.  What 
is  its  true  length  ? 


DIVISION  OF  FRACTIONS  165 

20.  I  had  a  sum  of  money,  of  which.  I  paid  away  ^,  then  i  of 
the  remainder,  and  found  that  I  had  still  left  $  46.  What  sum 
had  I  at  first  ? 

21.  A  man  who  owns  |^  of  a  mill  sells  f  of  his  share.  What 
fraction  of  the  mill  does  he  still  own?  Had  he  sold  f  of  the 
mill,  what  fraction  of  the  mill  would  he  still  have  owned  ? 


111.  (1)  A  man  earns  84^  a  day,  and  his  daily  expenses 
are  $  1|.  How  many  days  will  it  take  him  to  save  enough 
money  to  buy  a  bicycle  costing  1 58|  ? 

The  sum  saved  each  day  =  $4^-$l|  =  $2^f. 
.'.  the  number  of  days  =  58|  h-  2^  =  25. 

(2)  A  farm  of*  340  A.  was  divided  between  two  sons,  so 
that  I  of  the  youngest  son's  share  was  equal  to  |  of  the  eldest 
son's  share.     Find  the  size  of  each  farm. 

I  of  the  youngest  son's  share  =  f  of  the  eldest  son's  share. 

The  youngest  son's  share  =  f  -4- 1,  or  |  of  the  eldest  son's  share. 
•  ••  f  +  f  5  or  y  of  the  oldest  son's  share  =  340  A. 

.  •.  the  oldest  son's  share  =  340  A.  -^  y  =  180  A. 
. '.  the  youngest  son's  share  =  f  of  180  A.  =  160  A. 
Complete  the  following  solution  of  question  (2) : 
Let  f  of  the  youngest  son's  share  =  6  units. 

(3)  Sold  tea-  at  92^  per  pound,  having  gained  -^-^  of  the 
cost.     Find  the  selling  price  per  pound  if  he  had  lost  ^q. 

Let  the  cost  of  1  lb.  be  measured  by  20  units. 
Then  ^j^  of  the  cost  price  =    3  units. 
The  first  selling  price        =  23  units. 
The  second  selling  price   =17  units. 
.-.  the  second  selling  price   =  i|  of  the  first. 


166  ARITHMETIC 

Exercise  107 

1.  What  part  of  a  dollar  is  75^?     33^^?    62^?    87J^? 
6J^?    IGi^? 

2.  At    12i^   each,  how   many    pillow   cases  will   cost  $2? 
(2-i  =  ?) 

3.  I  paid  $18  for  cloth  to  make  a  dress,  at  $1.50  a  yard. 
Find  the  number  of  yards.     (18  -^  1^  =  ?) 

4.  At  33|^  P  a  pair,  how  many  pairs  of  stockings  will  cost  f  3  ? 

5.  At  $  2.75  a  pair,  how  many  pairs  of  curtains  will  cost  $  11  ? 

6.  At  37|^^  a  pair,  how  many  dozen  sheets  will  cost  $  18  ? 

7.  At  $3.75  a  pair,  how  many  pairs  of  curtains  will  cost 
$26.25? 

8.  At  3  for  50  ^,  how  many  handkerchiefs  will  cost  $  3.50  ? 

9.  At  $  10.50  each,  how  many  rugs  will  cost  $  42  ? 

10.  At  $  5.75  per  barrel,  how  many  barrels  of  flour  will  cost 
$51.75? 

11.  At  621^  a  bushel,  how  many  bushels  of  wheat  will  cost 
$250? 

12.  At  37|^^  a  bushel,  how  many  bushels  of  oats  will  cost  $  36  ? 

13.  How  many  cans,  each  containing  1^  qt.,  can  be  filled  from 
a  barrel  containing  31|  gal.  of  syrup  ? 

14.  The  bottom  of  a  box  measures  4  ft.  by  2^  ft.  How  deep 
must  it  be  to  contain  20  cu.  ft.  ? 

15.  The  bottom  of  an  excavation  measures  7  ft.  6  in.  by  3  ft. 
2  in.     How  deep  must  it  be  to  contain  95  cu.  ft.  of  water  ? 

16.  What  is  the  cost  price  of  goods  sold  for  $  1.20,  at  a  gain  of 
J  of  the  cost  ?  Of  I  of  the  cost  ?  Of  }-  of  the  cost  ?  Of  ^  of  the 
cost  ?     Of  Y^j  of  the  cost  ?    Find  the  gain  in  each  case. 

17.  A  man  sold  24  horses  for  $  150  each ;  on  half  of  them  he 
gained  \  o£  what  they  cost ;  and  on  the  remainder  he  lost  |  of 
what  they  cost.     Find  his  whole  gain  or  loss. 


DIVISION  OF  FRACTIONS  167 

18.  The  sum  paid  for  three  copies  of  a  certain  book,  including 
a  duty  of  J  of  the  cost,  was  $  2.25.  Find  the  original  cost  and 
the  duty  on  one  book. 

19.  The  sum  paid  for  494  gal.  of  oil,  including  a  duty  on  each 
gallon  which  amounted  to  ^  of  the  cost  price  of  a  gallon,  was 
$1719.12.     Find  the  duty  on  each  gallon. 

20.  By  selling  oranges  at  the  rate  of  $  2.60  for  4  doz.,  it  was 
found  that  f  of  the  cost  was  gained.  Find  the  price  at  which 
each  orange  ought  to  have  been  sold  in  order  to  gain  50  %  of  the 
cost. 

21.  A  person  who  has  |  of  a  mine  sells  f  of  his  share  for 
1 6000.      What  is  the  value  of  the  whole  mine  ? 

22.  A  grocer  in  selling  goods  sells  15J  oz.  for  1  lb.  How 
much  does  he  cheat  a  customer  who  buys  to  the  amount  of  $  40  ? 

23.  A  cannon  ball  travels  at  the  rate  of  1500  ft.  in  li  sec. 
How  far  will  it  have  gone  in  \^  of  a  minute  ? 

24.  Given,  that  pure  water  is  composed  of  oxygen  and  hydro- 
gen in  the  proportion  by  weight  of  15  to  2,  find  the  weight  of 
each  in  a  cubic  foot  of  water.  (A  cubic  foot  of  water  weighs 
1000  oz.) 

25.  A  tank  is  9  ft.  long,  3  ft.  4  in.  wide,  and  2  ft.  6  in.  deep. 
Find  the  number  of  cubic  feet  of  water  it  will  hold.  How  many 
gallons  ?     (1  cu.  ft.  =  7.48  gal.) 

26.  A  tank  is  6  ft.  8  in.  long,  2  ft.  6  in.  wide,  and  1  ft.  6  in. 
deep.     F'^ind  the  number  of  gallons  of  water  it  will  hold. 

27.  A  man  willed  ^  of  his  property  to  his  wife,  -|  of  the 
remainder  to  his  daughter,  and  the  rest  to  his  son.  The  differ- 
ence between  the  wife's  and  daughter's  shares  was  $  2500.  Find 
the  value  of  his  property. 

28.  A  grain  dealer  paid  61|  ^  a  bushel  for  corn,  and  sold  it  for 
C)0|  ^  a  bushel.  He  lost  $  375  on  the  transaction.  Find  the  num- 
ber of  bushels. 


168 


ARITHMETIC 


Miscellaneous  Exercise  108 

1.  The  citizens  of  Madison,  Wis.,  have  raised  the  following 
sums  of  money  by  private  subscription  for  public  improvements 
in  the  seven  years  preceding  1900 :  $  6888.86,  $  655,  f  995,  ^  1580, 
$10,160,  $2171,  $3231.50.     Find  the  total  amount. 

2.  The  follovi^ing  sums  of  money  were  collected  under  the 
War  Revenue  Act  from  June  13,  1898,  to  March  31,  1900,  cents 
omitted.     Find  the  total  amount  collected : 


Objects  of  Taxation 


Amount  Collected 


Cigars 

Cigarettes 

Snuff 

Tobacco,  chewing  and  smoking         .        .        .        . 

Dealers  in  leaf  tobacco 

Dealers  in  manufactured  tobacco 

Manufacturers  of  tobacco 

Manufacturers  of  cigars 

Miscellaneous  collections  relating  to  tobacco     . 

Fermented  liquors 

Additional  collections  on  fermented  liquors  stored   in 

warehouse 

Mixed  flour 

Bankers,  capital  not  exceeding  .$25,000    .... 
Bankers,  capital  exceeding  $25,000,  for  each  additional 

$1000  in  excess  of  $25,000 

Billiard  rooms 

Brokers,  stocks,  bonds,  etc.      .        .        . 

Brokers,  commercial 

Brokers,  custom  house 

Brokers,  pawn 

Bowling  alleys 

Circuses 

Exhibitions  not  otherwise  provided  for     .... 
Theatres,  museums,  and  concert  halls      .... 

Legacies 

Schedule  A 

Schedule  B 

Excise  tax  on  gross  receipts 


$5,202,691 

2,442,020 

1,641,281 

27,070,113 

127,170 

30,637 

39,193 

446,724 

773,175 

56,936,631 

197,936 

14,154 

712,426 

6,066,155 

683,443 

659,356 

277,016 

11,860 

71,776 

90,626 

28,929 

148,759 

97,729 

2,896,306 

66,781,776 

8,693,881 

1,463,547 


DIVISION  OF  FRACTIONS 


169 


3.  The  following  statement  shows  the  imports  from  and  the 
exports  to  the  Philippines,  including  gold  and  silver,  for  the  four 
months  ending  Oct.  31, 1899.     Find  the  total  imports  and  exports : 


Imports  fkom 

Exports  to 

United  States 

$635,495 

11,518,748 

Canada  . 

174 

None 

United  Kingdom 

1,515,893 

1,683,806 

Spain      . 

1,008,813 

541,416 

France    . 

112,944 

62,953 

Germany 

650,425 

19,961 

Belgium 

48,299 

None 

Gibraltar 

None 

252,044 

Netherlands 

119,597 

None 

Russia    . 

48,867 

None 

Switzerland 

77,470 

None 

Italy       . 

45,904 

1,720 

Denmark 

7,953 

None 

Austria  . 

15,281 

None 

Japan     . 

82,098 

486,324 

China     . 

4,165,949 

1,463,638 

British  East  Indies 

876,177 

193,594 

Dutch  East  Indies 

22,593 

7,384 

Australia 

424,380 

180,345 

German  Oceanica 

None 

693 

British  Africa 

None 

2,510 

4.  As  the  result  of  a  freight  rate  war  over  wool,  the  rate  from 
Denver  to  Boston  was  reduced  from  $  1.54  to  $  1.15  per  100  lb. 
Find  the  freight  saved  on  a  shipment  of  135,000  lb. 

5.  Wire  fencing  is  bought  at  ^4.90  a  bale  of  20  rd.,  and  sold 
at  2  ^  a  foot.     Find  the  gain  on  12  bales. 

6.  Find  the  value  of  6  beeves,  average  weight  1500  lb.,  at 
$5.85  per  100  1b. 

7.  The  Chicago  real  estate  sales  for  the  week  ending  Aug.  25, 
1899,  were  $  1,336,455,  and  for  the  corresponding  week  of  the 
preceding  year  were  f  889,709.     Find  the  increase. 


170 


ARITHMETIC 


8.  The  freight  on  grain  from  Chicago  to  New  York  is  15^ 
per  100  lb.     What  is  the  rate  per  bushel  ?     On  8000  bu.  ? 

9.  What  is  the  ratio  of  150  lb.  to  100  lb.     Find  the  selling 
price  of  a  hog  weighing  150  lb.  at  $  4.75  per  100  lb. 

10.  Find  the  selling  price  of  a  hog  weighing  180  lb.  at  $4.85 
per  100  lb. 

11.  Find  the  selling  price  of  50  hogs,  average  weight  120  lb., 
at  $4.15  per  100  1b. 

12.  Find  the  cost  of  a  190-lb.  lamb  at  $6  per  100  lb. 

13.  Find  the  cost  of  250  128-lb.  lambs  at  $6.20  per  100  lb. 

14.  The  record  of  the  temperature  at  Chicago  for  March  10-11, 
1900,  is  given  in  the  following  table.  Find  the  average  tempera- 
ture during  the  day : 


4  p.m. 

5  p.m. 

6  P.M. 

7  P.M. 

8  P.M. 

9  p.m. 

10  P.M.   . 

11  P.M. 

12  midnighl 

1  A.M. 

2  A.M. 

3  A.M. 

.  42 
.  44 
.  42 
.  40 
.  34 
.  33 
.  32 
.  32 
.  32 
.  32 
.  33 
.  32 

4  A.M. 

5  A.M. 

6  A.M. 

7  A.M. 

8  A.M. 

9  A.M. 

10  A.M. 

11  A.M. 

12  M.    . 

1  P.M. 

2  P.M. 

3  P.M. 


32 
32 
32 
32 
34 
34 
34 
35 
38 
39 
39 
39 


15.  The  rate  of  insurance  on  a  building,  which  was  insured 
for  $  25,000,  was  advanced  from  1 1.85  to  $  2.34  per  $  100  insured. 
Find  the  increase  in  the  cost  of  insuring  the  building. 


CHAPTER  XII 

DECIMALS  * 

f 

Exercise  109 

1.  With  a  metric  stick  measure  a  distance  2  meters  (2  m.)  long ; 
1  meter  6  decimeters  (1  m.  6  dm.) ;  6  decimeters  5  centimeters 
(6  dm.  5  cm.)  ;   5  centimeters  8  millimeters  (5  cm.  8  mm.). 

2.  Measure  the  following  distances : 

1  m.    6  dm.  4  dm.  8  cm.  1  m.    2  dm.  4  cm. 

2  m.    4  dm.  6  cm.  5  mm.  2  dm.  4  cm.  5  mm. 

9  dm.  1  m.    9  cm.  6  dm.  1  cm.  7  mm. 

5  dm.  6  cm.  4  dm.  8  mm.  1  m.    4  dm.  6  cm. 

3.  How  many  decimeters  in  1  m.  ?  How  many  centimeters  in 
1  m.  ?     How  many  millimeters  in  1  m.  ? 

4.  Measure  the  following  distances : 

1.2  m.  .54  m.  .265  m.  .04  m. 

1.8  m.  .73  m.  .492  m.  .045  m. 

.6  m.  .475  m.  .309  m.  .005  m. 

5.  Read  the  following  expressions  :  4.56  m.  (four  and  fifty-six 
hundredths  meters),  2.35  m.,  1.64  mi.,  4.25  yr.,  .28  da.,  .16  hr., 
.08  min.,  2.425  lb.,  6.405  bu.,  .345  yd.,  .248  yd.,  .049  ft.,  .009  in., 
.006  in.,  2.243,  .065,  .06,  .005. 

6.  Give  the  place  value  of  each  figure  in  the  preceding  example, 
thus:  2.45  m.  =  two  meters,  four-tenths  of  a  meter,  five-hundredths 
of  a  meter. 

*  If  the  teacher  has  not  a  metric  stick,  the  work  in  decimals  may  be  based 
on  dollars,  cents,  and  mills. 

171 


172  ARITHMETIC 

7.  Measure  these  distances:  1.2  m.,  .34  m.,  .48  m.,  .425  m,, 
.369  m.,  .04  m.,  .08  m.,  .005  m.,  .009  m. 

112.  The  standard  unit  of  money  is  $  1.  One  dime  is  one- 
tenth  of  f  1,  and  1  ^  is  one-tenth  of  1  dime.  Hence  we  write 
$1,  1  dime,  and  1^  thus,  using  the  dollar  sign;  $1.11. 

Write  in  terms  of  1 1,  the  sum  of : 

(1)  $1,  2  dimes,  and  4^. 

(2)  1  ten-dollar  bill,  $1,  1  dime,  and  1^. 

(3)  1  one-hundred-dollar  bill,  1  ten-dollar  bill,  1 1, 1  dime, 
and  1  ^. 

113.  In  the  metric  system  of  measures,  the  standard  unit 
is  1  meter. 

One  decimeter  is  one-tenth  of  1  meter,  1  centimeter  is  one- 
tenth  of  1  decimeter,  and  1  millimeter  is  one-tenth  of  1  centi- 
meter. Since  the  same  relation  holds  between  these  units 
as  between  1  dime  and  $1,1^  and  1  dime,  we  can  express  the 
results  of  measurements  in  terms  of  1  meter,  as  we  express 
money  in  terms  of  $1. 

Hence  if  we  measure  with  the  metric  stick  a  distance  equal 
to  1  meter,  1  decimeter,  and  1  centimeter,  we  can  express  the 
distance  thus  :  1.11  meters. 

Measure  the  following  distances  and  express  them  in 
meters : 

(1)  2  meters,  8  decimeters,  and  4  centimeters. 

(2)  1  meter,  2  decimeters,  3  centimeters,  and  5  millimeters. 

(3)  1  meter,  1  decimeter,  1  centimeter,  and  1  millimeter. 

114.  Notation  and  Numeration.  —  Consider  the  num- 
ber 111 :  the  first  1,  beginning  at  the  right,  denotes  one  unit ; 
the  second,  one  ten  or  ten  units ;  the  third,  one  hundred  or 
one  hundred  units. 


DECIMALS  173 

The  third  1  is  equivalent  to  one  hundred  times  the  first  1, 
and  to  ten  times  the  second  1 ;  the  second  is  equivalent  to 
ten  times  the  first  1,  and  to  one-tenth  of  the  third  1 ;  the 
first  1  is  equivalent  to  one-tenth  of  the  second  1,  and  to  one- 
hundredth  of  the  third  1. 

Now  rewrite  the  number  111,  place  a  point  after  the  first 
1  to  indicate  that  this  1  is  to  be  regarded  as  representing  the 
standard  unit,  and  then  place  after  the  point  three  I's,  so 

that  we  have 

111.111. 

We  may  ask  what  each  of  these  I's  should  mean,  if  the 
same  relation  is  to  hold  among  successive  digits  that  we 
have  supposed  hitherto  to  hold.  The  1  after  the  point  would 
naturally  mean  one  tenth.  The  next  1  to  the  right  would 
naturally  mean  one  hundredth.  It  is  one-tenth  of  the  pre- 
ceding one  —  that  is,  one-tenth  of  one  tenth. 

Similarly,  the  next  1  would  signify  one  thousandth,  and 
would  equal  one-hundredth  of  the  one  tenth  or  one-tenth 
of  the  one  hundredth.  Thus,  the  number  111.111  may  be 
written  as  follows  :  One  hundred,  one  ten,  one  unit,  one 
tenth,  one  hundredth,  and  one  thousandth. 

Again,  the  one  to  the  extreme  right  is  1  thousandth :  the 
next  1  is,  from  its  position,  equivalent  to  10  thousandths ; 
and  the  next  1  is  100  thousandths.  So  that  to  the  right  of 
the  point  we  have  111  thousandths.  The  whole  number  may 
now  be  read  one  hundred  eleven,  and  one  hundred  eleven 
thousandths. 

115.  A  Decimal  Fraction  or  a  decimal  is  one  which  has  for 
its  denominator  10,  100,  1000,  or  some  power  of  10. 

The  Power  of  a  number  is  the  product  found  by  multiply- 
ing the  number  by  itself  one  or  more  times ;  thus,  100  or  10^ 
is  the  second  power  of  10  ;  1000  or  10^,  the  third  power  of  10. 


174  ARITHMETIC 

The  denominator  of  a  decimal  fraction  is  never  expressed  ; 
thus  ^  and  -f^^  are  written  as  decimals,  .5  and  .57. 

The  point  placed  to  the  right  of  the  one-unit  and  between 
it  and  the  tenth-unit  is  called  the  decimal  point. 

116.    To  change  a  decimal  to  a  common  fraction  in  its  lowest 

terms. 

.12  =  i^  =  i\ ;  .425  =  ^^  =  2'\Aj  =  H- 

2.76  =  2^^  =  2| ;  6.036  =  6^^  =  6^^. 

Conversely,  S^^^^^  =  5.29  ;      92^^  =  92.025. 

Exercise  110 

1.  Eead  the  numbers  in  Exercises  111  and  112,  expressing 
them  in  terms  of  different  units,  as  1  m.,  1  mi.,  1  lb. 

2.  Kead.5;  .05;  .005;  .0005;  .004;  .075 ;  2.008 ;  3.029 ;  .0006, 

3.  Eead  the  following  expressions  :  2.548  T. ;  2.917  A. ;  4.73  A. ; 
8.04  mi.;  3.04  yr;;  4.007  da.;  .346  mo. ;  .85hr.;  .008  bu. ;  .0009  T.; 
4.444;  5.032;  6.2059;  8.0063;  7.2906;  .043. 

4.  Read  the  place  value  of  each  figure  in  the  previous  example. 

Eeduce  the  following  decimals  to  common  fractions  in  their 
lowest  terms : 

5.  .5;  .8;  .75;  .45;  .72;  .05;  .125;  .625. 

6.  2.4;  7.25;  4.375:  3.875;  3.0875;  6.325. 

7.  2.125  T.;  ^.0625;  8.225  A. ;  ^9.375. 

8.  Find  the  cost  of : 

12  bars  of  soap  at  6.25^  a  bar. 
16  lb.  sugar  at  5.125^  a  pound. 
2  doz.  pairs  rubbers  at  $  .375  a  pair. 
64  sheep  at  $4,875  each. 

9.  A  farmer  gave  .875  of  his  farm  of  400  A.  to  his  three  sons. 
How  many  acres  did  he  keep  for  himself  ? 


ADDITION  OF  DECIMALS  175 

10.  A  merchant  sold  cloth  at  an  advance  of  .375  of  the  cost, 
and  gained  24  ^  a  yard.  "What  did  his  customers  pay  a  yard  for 
the  cloth  ? 

Express  in  figures : 

11.  Four  tenths  ;  two,  and  fifty-six  hundredths ;  three,  and  five 
hundredths ;  sixty-six  hundredths ;  six  hundredths ;  three  hun- 
dred sixty-eight  thousandths ;  seventy-nine  thousandths ;  eight 
thousandths. 

12.  Seventy-five,  and  twenty-eight  thousandths  ;  three  hundred 
twenty-nine,  and  ninety-four  thousandths;  two,  and  five  hun- 
dredths ;  seven,  and  seven  thousandths ;  two  hundred,  and  two 
hundredths  ;  two  thousand,  and  two  thousandths. 

13.  Five  thousand  nine  hundred  forty-two  ten-thousandths; 
eight  hundred  seventy-one  ten-thousandths;  forty-five  ten-thou- 
sandths ;  six  ten-thousandths. 

14.  Express  the  following  fractions  as  decimals:  -j^;  -f-^;  y%\; 

6.      243.        29.         7       .Q8.A3.     AQ_.5  4      .     7^  3_1  8__9_  .     A      5 JJ        . 
To7?    TO'O^J    TTT'OU  J    TOOT  5    ^TTT?    "TOO  5    ^^  1'^'^^  ^     '"lOT^OOJ    "lUWCf  J 

-■- '  TO  OU- 

ADDITION  OF  DECIMALS 
117.   What  is  the  sum  of  4.9,  6.084,  24.32,  and  .8976  ? 

^•^  In  arranging  the  numbers,  be  careful  to  put  the  decimal 

6.084  points  directly  under  each  other,  thus  bringing  units  under 

24,32  units,  tenths  under  tenths,  etc.     Then  begin  at  the  lowest 

.8976  order  and  add  as  if  the  figures  were  integers,  putting  the 

36  2016  decimal  between  the  unit  and  the  tenths'  place. 

Exercise  111 

Find  the  sum  and  prove  your  answers  correct  by  adding  down 
the  columns : 

1.     53.674  2.       9.72  3.         .8592 

4.009  20.492  913.7451 

821.646  .0487  21.0106 

2.182  918.0006  47.9 


176  ARITHMETIC 

4.  Explain  step  by  step  the  process  of  addition  in  the  first 
three  examples. 

Write  in  columns  and  add: 

5.  6.5  +  32.47  +  2.048  +  59. 

6.  .452 +4.08 +  .646 +  .06 +  49.027. 

7.  4.0406  +  213.939  +  2.91  +  3.04. 

8.  89432.1  +  7.65439  +  .0084  +  8400. 

9.  27.064  +  .0012  +  394.2001  +  .819. 

10.  How  many  yards  are  there  in  five  pieces  of  cloth,  the  first 
of  which  contains  37.5  yd.,  the  second  26.75,  the  third  14.375,  the 
fourth  36.5,  and  the  .fifth  63.125  ? 

11.  Four  sections  of  land  contain  the  following  areas:  24.729 
sq.  mi.,  92.04  sq.  mi.,  8.007  sq.  mi.,  and  36.429  sq.  mi.  Find  the 
total  area. 

12.  Find  the  sum  of  sixty-one  ten-thousandths;  eight,  and 
seven  thousand  six  hundred  ninety-five  ten-thousandths;  nine 
thousand  seven  hundred  eighty-six  ten-thousandths. 

13.  During  the  year  1899,  in  the  city  of  Chicago,  11.32  mi.  of 
streets  were  paved  with  asphalt,  10.17  mi.  with  cedar,  13.07  mi. 
with  brick,  1.59  mi.  with  granite,  and  8.09  mi.  with  macadam. 
Find  the  total. 

14.  The  north  side  of  the  city  of  Chicago  has  parks  containing 
the  following  number  of  acres  :  320  A.,  2.3  A.,  .53  A.,  .2  A.,  .19  A., 
.46  A.,  .0225  A.,  .0482  A.     Find  their  total  area. 

15.  The  west  side  of  the  city  of  Chicago  has  parks  containing 
the  following  number  of  acres :  200.62  A.,  185.87  A.,  179.79  A., 
14.8  A.,  15.79  A.,  5.42  A.,  4.51  A.,  4.89  A.,  5.5  A.,  6.06  A.,  3.65  A., 
.94  A.,  .25  A.,  2.38  A.,  .68  A.,  1.08  A.,  2  A.,  1.28  A.  Find  their 
total  area. 

16.  A  man  paid  a  state  tax  of  $  .45,  a  city  tax  of  $  1.12,  and  a 
school  tax  of  $  1.78  on  every  f  100  property.  Find  his  total  tax 
on  each  $  100  property. 


SUBTRACTION  OF  DECIMALS 


177 


17.    The  tax  rates  for  the  west  town  of  the  city  of  Chicago  for 
1899  and  1898  were  for  each  $  100  property  as  follows : 


1899 

1898 

State      .... 

$0.42 

$0.56 

County  .... 

.548 

.78 

Town     .... 

.035 

.29 

City 

L27 

2.65 

Library  .... 

.049 

.06 

Sanitary . 
Park  .     . 
Boulevard 
Town  bond 
School     . 


1899 


$.893 
.847 
.05 
.059 

L829 


1898 


$L50 

LIO 

.05 

.10 

2.77 


Find  the  total  tax  on  $  100  property. 


SUBTRACTION  OF  DECIMALS 
118.   From  29.364  take  3.87049. 


29.36400 
3.87049 


29.364 
3.87049 


As  in  addition  of  decimals,  place  the  decimal  points  under  each  other, 
thus  placing  units  under  units,  tenths  under  tenths,  etc. 

As  the  value  of  the  decimal  is  not  changed  by  annexing  zeros  to  the  right 
of  the  decimal,  annex  in  this  case  two  zeros.  Subtract  as  in  whole  numbers, 
and  place  the  decimal  point  in  the  remainder  between  the  unit  and  the  tenths' 
place. 


From 

1.   8.43 

Exercise  11 
2.    13.47016 

2 
3. 

.503 

4. 

.52 

Take 

2.95 

2.0984 

.28914 

.13064 

5.  Explain  step  by  step  the  process  of  subtraction  in  the  first 
three  examples. 

Find  the  difference  and  prove  your  answers  correct : 

6.  .62 -.47.           10.    .07 -.059.  13.    .7304  -  .67. 

7.  .73 -.35.          11.   8.9-3.4265.  14.   4.8295-3.9998. 

8.  .894 -.406.       12.   39.42-15.9879.  15.   2.03  -  .00428. 

9.  .74 -.365. 


178  ARITHMETIC 

16.  The  parks  on  the  north  side  of  Chicago  contain  323.7507  A., 
and  those  on  the  west  side  615.45  A.  How  many  more  acres  of 
parks  on  the  west  than  on  the  north  side  of  the  city  ? 

17.  The  height  of  the  barometer  at  Davenport,  la.,  May  3, 
1900,  was  30.96.  How  much  greater  is  this  than  the  height  at 
each  of  the  following  places  ? 

Albany  .     .  29.62        Nashville  .     .  30.04        Buffalo  .     .  29.82 
Cleveland   .  29.96        Montreal    .     .  29.54        Oklahoma  .  30.16 

18.  Explain  whether  .067  or  .068  is  nearer  .06748,  and  express 
in  words  the  difference  in  each  case. 

19.  The  length  of  a  seconds  pendulum  is  39.1392  in.,  and  that 
of  a  meter  is  39.371  in.     Find  the  difference  in  their  lengths. 

20.  Find  the  difference  between  the  length  of  1  meter  and 
1  yard. 

21.  From  a  i)iece  of  cloth  containing  35.5  yd.,  a  merchant  sold 
12.75  yd.     How  much  was  left  ? 

22.  Find  the  difference  between  $  11  A%  and  35  ^. 

23.  Monday  the  barometer  read  29.86,  Tuesday  it  rose  .12,  and 
Wednesday  it  fell  .28.  What  was  the  height  of  the  barometer 
Wednesday  evening  ? 

24.  The  mercury  in  a  barometer  rose  .121  in.,  .073  in.,  and 
.019  in.  in  three  successive  days;  it  fell  .054  in.  and  .065  in. 
during  the  two  following  days,  rose  .053  in.  on  the  sixth  day,  and 
fell  .028  in.  on  the  seventh  day.  If  its  height  at  the  beginning 
of  the  first  day  was  30.078  in.,  what  was  its  height  at  the  close  of 
the  seventh  day  ? 

MULTIPLICATION  OF   DECIMALS 

119.  (1)  -^  multiplied  by  10  =  7,  and  therefore  .7  multi- 
plied by  10  =  7. 

f^^  multiplied  by  10  =  ^^^,  or  62.7,  and  therefore  6.27 
multiplied  by  10  =  62.7. 


MULTIPLICATION  OF   DECIMALS  179 

(2)  -^-f^  multiplied  by  100  =  75,  and  therefore  .75  multi- 
plied by  100  =  75. 

f fJI  multiplied  by  100  =  m^  =  627.5,  and  therefore  6.275 
multiplied  by  100  =  627.5. 

(3)  f  f^ff  multiplied  by  1000  =  6  2_iA8.  ^  6275.8,  and  there- 
fore 6.2758  multiplied  by  1000  =  6275.8. 

In  the  preceding  examples  how  many  places  to  the  right  was  the  decimal 
point  moved  on  multiplying  by  10  ?  100  ?  1000  ? 

Exercise  113 

1.  Multiply  each  of  the  following  numbers  by  10 : 

.6;  .8;  .84;  .95;  .842;  .763;  1.75;  2.439;  20.4. 

2.  State  how  to  multiply  a  decimal  by  10,  without  actually 
doing  the  work  of  multiplication. 

3.  Multiply  by  10  :  .06 ;  .04 ;  .005 ;  .0123 ;  .0044 ;  .0001. 

4.  Multiply  by  10:  42.3;  5.69;  .478;  54.793;  2.9342. 

5.  Multiply  by  100 :  .84.  9.65;  .763;  .003;  .04;  .246. 

6.  State  how  to  multiply  a  decimal  by  100,  without  actually 
doing  the  work  of  multiplication  ;  by  1000. 

7.  Eeduce    to    pounds:    2.34   cwt. ;    6.42   cwt. ;    .345    cwt. ; 
.125  cwt. ;  .24  cwt. ;  .06  cwt. ;  3.46  cwt. ;  2.468  cwt. 

8.  Find  the  number  of  pounds  in  1.24  cwt.  of  sugar.     Find  its 
value  at  6  j^  a  pound. 

9.  Multiply  by  1000:  .982;  .0642;  .0009;  .008;  .0123;  .0004. 

10.  Multiply  by  100:  .86;  SS-,  .9;  .060;  9.8;  .4;  6.245;  .005. 

11.  Multiply  by  1000:  .594;  5.94;  59.4;  .007;  .07;  .7;  3.14;  2.5. 
.  12.    State  how  to  divide  a  decimal  by  10.     By  100.     By  1000. 

13.  Divide  by  10:  27;  82.19;  4.8;  52.93;  .4;  .06;  .009. 

14.  Divide  by  100:  482;  76;  415.62;  8.1;  .78;  .4;  .09;  789.46. 

15.  Eeduce   to   hundredweight:    400   lb.;    237   lb.;    575   lb.; 
832  lb. ;  3475  lb. ;  4759  lb. ;  67  lb. ;  25  lb. ;  87  lb. ;  95  lb. 


180 


ARITHMETIC 


16.  69  lb.  =  ?  cwt.  Find  the  cost  of  a  69-lb.  lamb  at  $5  per 
hundredweight. 

17.  Find  the  number  of  hundredweight  in  16  85-lb.  lambs;  in 
24  97-lb.  sheep. 

18.  Divide  by  1000:  643;  2459.7;  .69;  2.31;  .03;  6009;  17643. 

19.  Divide  8436  by  1000;  by  2000. 

20.  Divide  by  2000:  6436;  2974;  3298;  7935;  428;  526;  439; 
24;  56;  77. 

21.  Reduce  to  tons:  5436  lb.;  3138  lb.;  624  lb.;  7296  lb.; 
518  lb. ;  14324  lb. 

22.  Find  the  number  of  tons  in  a  carload  of  coal  weighing 
39,374  lb. 

23.  During  the  ten  years  preceding  July,  1899,  the  wholesale 
price  of  bacon  per  pound  decreased  from  $  .056  to  $  .052.  Find 
the  decrease  in  the  wholesale  price  of  1  lb. ;  of  1000  lb. 

24.  The  average  wholesale  price  of  the  following  articles,  Jan- 
uary, 1890,  and  July,  1899,  are  given  below.  Find  the  difference 
in  price  on  an  order  of  1000  lb.  in  each  case. 


July,  1899 


Lard  per  pound 

Ham  per  pound 

Sugar,  granulated,  per  pound 

Thread,  spool 

Starch,  silver  gloss,  per  pound 


$.049 
.1076 
.052 
.031 
.058 


25.  If  sheep  are  quoted  at  $5.25  per  hundredweight  {i.e.  per 
100  lb.),  find  the  price  per  pound  as  a  decimal  of  a  dollar. 

26.  Live  stock  is  quoted  at  the  following  prices  per  hundred- 
weight. Find  in  each  case  the  price  per  pound  as  a  decimal  of  a 
dollar:  $5.45;  $2.25;  $4.10;  $4.50;  $8.75;  $5.37^;  $5.42^; 
$7.10;  $5.32 J. 

27.  A  sheep  weighs  87  lb.  Express  this  as  a  decimal  of  100  lb.,  i.e. 
of  1  cwt.    A  cow  weighs  769  lb.    This  is  how  many  hundredweight  ? 


MULTIPLICATION  OF   DECIMALS  181 

28.  How  many  hundredweight  will  12  89-lb.  Texas  sheep 
weigh  ?     Find  their  value  at  $  5  per  hundredweight. 

29.  How  many  thousand  shingles  in  5000?  6200?  4375? 
9360  ?     478  ? 

30.  How  many  tons  of  coal  in  6000  lb.?  4800  lb.?  3468  lb.? 
7296  lb.  ? 

31.  Write  as  decimals:  4%;  6%;  2^%;  4i%;  8%;  5}%;  6i%; 
31%;  9%;  37%;  25%;  23i%  ;  16.4%;  27.6%;  8.2%;  18|%. 

120.  (1)    Multiply  6.24  by  46. 

6.24 

46 

37.44 
249.6 

287.04 

Here  we  multiply  4  hundredths  by  6,  and  the  product  is  24  hundredths, 
or  2  tenths  4  hundredths.  Again,  we  multiply  2  tenths  by  6,  and,  adding  in 
the  2  tenths,  the  result  is  14  tenths,  or  1  unit  4  tenths,  and  so  on.  Next, 
multiplying  by  4,  we  must  write  the  results  one  place  to  the  left,  as  in  the 
multiplication  of  integers.  In  multiplying,  it  is  as  well  to  omit  the  decimal 
points  from  the  partial  products  37.44  and  249.6. 

(2)    Multiply  6.24  by  4.6. 

6.24 

4.6 

3744 

2496 

28.704 

This  differs  from  the  former  question  only  in  that  the  multiplier  4.6  is 
one-tenth  of  46,  and  therefore  the  product  is  one-tenth  as  large,  or  28. 704. 

121.  From  this  and  other  similar  problems  the  rule  can  be 
deduced : 

To  multiply  two  decimals^  proceed  as  if  they  were  integers^ 
and  mark  off  in  the  product  as  many  decimal  places  as  there 
are  in  both  the  multiplier  and  the  multiplicand. 


182  ARITHMETIC 

122.    (1)    Multiply  2.56  by  .94. 

2.56 
.94 
1024 
2304 


2.4064 

(2)   Multiply  .249  by  .035. 

.249 

.035 

.    .  1245 

747 


.008715 
Explanation.  — 

.249  X  .035  =  ,%V^  X  tMtt  =  ii^VVW  =  -008715. 
123.   What  is  the  interest  *  on  i  468  for  1  yr.  at  6  ^  ? 

$  468  principal 
$  .06  rate  per  unit 
$28.08  interest  for  1  yr. 

.-.  The  interest  on  $468  for  1  yr.  at  6%  =  $28.08.    To  find  the  interest 
for  6  mo.  take  .03  of  $468,  or  divide  §28.08  by  2. 


Ml 

iltiply: 

Exercise 

114 

1. 

3.26 

5.29 

3.148 

.4378 

.8664 

.0581 

4 

.6 

.9 

.6 

.5 

.8 

2. 

58.9 

.685 

.072 

.093 

.182 

2.98 

2.4 

.36 

.41 

M 

.75 

3.5 

3. 

3.1416 

59.38 

.0075 

.094 

.006 

9271 

.72 

3.12 

9.8 

.04 

.006 

.004 

See  §  207  for  definitions  of  principal  and  interest 


MULTIPLICATION  OF  DECIMALS  183 

Find  the  interest  for  1  yr.  on : 

4.  f26at  5%;  $248at  6%;  $16  at  4%. 

5.  $46.50  at  8%;  1894.75  at  3%;  $2389.20  at  7%. 

6.  $324  at  21%;  $704.60  at  3i%;  $852.94  at  4i%. 

Find  the  interest  on : 

7.  $237  for  6  mo.  at  6%;  $628  for  3  mo.  at  8%. 

8.  $22.58  for  8  mo.  at  6%;  $69.50  for  9  mo.  at  8%. 

9.  $6438  for  4  mo.  at  1\%  ;  $523.60  for  8  mo.  at  7^% ;  $250 
for  10  mo.  at  6%;  $495.80  for  1  mo.  at  6%. 

10.  Find  .375  of  45  mi.;  .0625  of  640  A.;  .0875  of  $415.60. 

Multiply : 

11.  4.8  X  5.12;  .21  x  4.67.  16.  .8  x  .8;  .09  x  .09. 

12.  3.1416  X  .02  ;  1.46  x  .39.  17.  .1  X  .1 ;  .01  x  .01. 

13.  .004  X  99;  .004  x  .005.  18.  .2  x  .2;  .7  x  .7 

14.  $249  X  1.04.  19.  .375  x  2.15;  .0375  x  2.15. 

15.  .84  X  .251;  2.04  x  .0037.  20.  .051  x  .042;  .014  x  .0038. 

21.  I  sold  an  article  which  cost  me  $265  at  a  gain  of  23%. 
Find  the  gain.     Find  the  selling  price. 

22.  I  sold  an  article  that  cost  $226.50  at  a  loss  of  34%.  Find 
the  loss  and  the  selling  price. 

23.  Measure  the  diameter  of  some  circular  object,  and  also  the 
circumference.  Multiply  the  diameter  by  3.1416,  in  which  case 
the  product  should  equal  the  circumference. 

24.  To  find  the  ciixumference  of  a  circle  multiply  the  diameter  by 
3.1416. 

25.  Find  the  circumferences  of  the  circles  whose  diameters  are : 

8  in.  2.8  in.  9.21  in.  .98  in. 

9  in.  3.4  in.  4.37  in.  .45  in. 
10  in.                      7.2  in.                      2.06  in.  .75  in. 


184  ARITHMETIC 

26.  Find  the  number  of  rods  of  barbed  wire  required  to  enclose 
a  circular  pond  whose  diameter  is  5  rd.,  there  being  4  rows  of  wire 
in  the  fence. 

27.  Find  the  cost  of  the  woven  wire  for  fencing  in  a  circular 
pond  10  rd.  in  diameter,  the  fencing  costing  $  6.72  per  bale  of  20  rd. 

Exercise  116 

1.  Keduce  to  inches :  .62  ft.;  .75  ft.;  .375  ft.;  .245  ft. 

2.  Keduce  to  feet:  .85yd.;  .46yd.;  .7  yd.;  .09yd.;  .768yd. 

3.  Reduce  to  yards  (multiply  by  5.5)  :  .84  rd.;  .72  rd.;  .456  rd. 
.324  rd.;  .07  rd. 

4.  Find  the  cost  of  .8  rd.  wire  cloth  at  $  .20  a  yard. 

5.  Reduce  to  feet  (multiply  by  16.5):  .4  rd.;  .96  rd.;  .44  rd.; 
.25  rd.;  .375  rd.;  .225  rd. 

6.  Find  the  cost  of  .6  rd.  wire  fencing  at  $  .20  a  foot. 

7.  Multiply  ^240  by  1.05  twice  in  succession. 

8.  Multiply  ^325  by  1.06  twice  in  succession. 

9.  Multiply  $415.80  by  1.04  and  the  result  by  1.02. 

10.  Multiply  $75  by  1.045  twice  in  succession. 

11.  Multiply  $975  by  1.065  twice  in  succession. 

12.  Multiply  3.1416  by  8;  by  7.5  ;  by  .04. 

13.  Multiply  7.48  by  9.1;  by  .04;  by  .006. 

14.  Multiply  the  square  of  5  by  3.1416. 

15.  A  railroad  engineer  gets  3.85^  per  mile.  How  much  will 
he  earn  in  a  month  if  his  runs  amount  to  3000  mi.? 

16.  One  meter  =  39.371  in.  Find  the  number  of  inches  in  a 
distance  equal  to  4  m. 

17.  The  length  of  a  wall,  according  to  the  French  metric 
system,  is  9.48  meters.  Find  its  length  in  inches,  the  length  of 
1  meter  being  39.371  in. 


MULTIPLICATION   OF   DECIMALS  185 

18.  Multiply  the  sum  of  2.616,  .00132,  and  1.0448  by  .626. 

19.  Find  the  length  of  the  fence  enclosing  an  oblong  garden 
2.5  rd.  long  and  1.5  rd.  wide.  What  will  it  cost  to  fence  it  at 
50^  a  rod? 

20.  A  piece  of  land  is  63.5  rd.  long  and  27.75  rd.  wide.  What 
will  it  cost  to  fence  it  at  $  .875  per  rod  ? 

21.  A  person  sold  .15  of  an  estate  to  one  person,  and  then  -^  of 
the  remainder  to  another  person.  What  part  of  the  estate  did  he 
still  retain  ? 

22.  If  a  business  produces  an  annual  return  of  $  12,000,  and  of 
three  partners  one  has  .465  and  another  .28  share  of  the  profits, 
how  much  money  falls  to  the  share  of  the  third  partner  ? 

23.  A  merchant  sells  28.5  yd.  of  cloth  which  cost  him  25j^  a 
yard,  for  37.5^  a  yard.     What  was  his  gain  ? 

24.  Find  the  weight  of  25  lambs,  averaging  65  lb.  each.  How 
many  hundredweight?  Find  their  value  at  $5.75  per  hundred- 
weight. 

25.  Find  the  value  of: 

30  109-lb.  lambs  at  $  6.25  per  hundredweight. 
27  122-lb.  sheep  at  $5.60  per  hundredweight. 
75  125-lb.  pigs  at  $  4.80  per  hundredweight. 
400  170-lb.  hogs  at  f  5.161  per  hundredweight. 
16  185-lb.  hogs  at  $5,421  per  hundredweight. 
11  1330-lb.  beeves  at  $4.90  per  hundredweight. 
19  1450-lb.  beeves  at  $  5.60  per  hundredweight. 

26.  Find  the  cost  of  6350  shingles  at  $2.80  per  1000. 

27.  How  many  tons  of  coal  in  6240  lb.  (write  as  a  decimal)? 
Find  its  value  at  $  6.25  a  ton. 

28.  Find  the  value  of  8466  lb.  of  coal  at  $ 5.75  a  ton;  of  744  lb. 
at  $6.50  a  ton. 

29.  Find  the  cost  of  976  lb.  hay  at  $11.50  a  ton. 


186  ARITHMETIC 

DIVISION  OF  DECIMALS 
124.    (1)  Reduce  19.2  in.  to  feet. 

19.2  in.  =  19.2  --  12  or  1.6  ft. 

(2)  May  30, 1900,  the  Philadelphia  baseball  team  had  won 
19  games  and  lost  10.     Find  the  percentage  of  games  won. 

The  total  number  of  games      =  19  +  10  =  29. 
The  percentage  of  games  won  =  19  -^  29  =  .655. 

.655 
29)19.000 
174 
160 
146 
160 
146 
5 

Exercise  116 

1.  Reduce  to  feet:    13.2  in.;    56.4   in.;    4.32   in.;    5.4   in. 
39.84  in. ;   274.08  in. ;   3.2412  in. 

2.  Reduce   to  yards:    2.4   ft.;    1.68  ft;    4.32   ft.;    10.8   ft. 
.48  ft.;   241.35  ft. 

3.  Reduce  to  yards:  19.44  in.;  75.24  in. ;  17.28  in.;  247.68  in. 
15.48  in. 

4.  Reduce  to  gallons  :  24.8  qt. ;  1.732  qt. ;  3.456  qt. ;  .712  qt. 
.312  qt. ;  .516  qt. 

5.  Reduce  to  weeks:  34.3  da.;  5.39  da.;  .245  da.;  29.12  da. 
456.4  da. 

6.  Reduce  to  days :    19.2  hr. ;    3.84  hr. ;    736.8  hr. ;    .456  hr. 
196.8  hr. 

7.  Reduce  to  bushels  :  22.4  qt. ;  4.16  qt. ;  8.32  qt. ;  238.08  qt. 
2.848  qt. 

8.  How  many  cubic  inches  in  1  cu.  ft.  ?     In  1  gal.  ? 

9.  Show  by  dividing  231  cu.  in.  into  1728  cu.  in.  that  1  cu.  ft 
=  7.48  gal.,  nearly. 


DIVISION  OF   DECIMALS 


187 


10.  How  many  gallons  in  3  cu.  ft.  ?  8  cu.  ft.  ?  15  cii.  ft.  ? 
36  cu.  ft.  ?     160  cu.  ft.  ?     1  cu.  ft.  =  7.48  gal. 

11.  A  cistern  contains  720  cu.  ft.  How  many  gallons  of  water 
will  it  hold  ? 

12.  A  baseball  team  won  29  games  and  lost  26.  Find  the  per- 
centage of  games  won. 

13.  The  record  of  the  following  baseball  teams  on  June  25, 
1900,  is  given  below.  Find  in  each  case  the  percentage  of  games 
won: 


Won 

Lost 

Won 

Lost 

Brooklyn 

Philadelphia    .... 

Pittsburg 

Boston 

33 
32 
25 
23 

17 
19 
27 
25 

Chicago 

Cincinnati      .... 

St.  Louis 

New  York 

24 
22 
20 
19 

27 
27 
27 
29 

125. 


8)24 
3 


8)2.4 
.3 


8). 24 
.03 


That  is,  24  divided  by  8  =  3. 

24  tenths  divided  by  8  =  3  tenths  =  .3. 

24  hundredths  divided  by  8  =  3  hundredths  : 

5.47 


.03. 


6).0018 
.0003 


46)251.62 
230 
216 
184 
322 
322 

That  is,  18  ten-thousandths  -=-6  =  3  ten-thousandths  =  .0003  ;  and  25,162 
hundredths  -=-  46  =  547  hundredths  =  5.47. 

50)150       500)1500       5000)15000 
3  3  3 


5)15 
3 


Therefore,  if  we  multiply  both  divisor  and  dividend  by  10, 
100,  1000,  and  so  on,  the  quotient  remains  unchanged. 


188  ARITHMI^TIC 

126.    (1)  Find  the  value  of  4.1262  -^  .69. 

The  quotient  of  4.1262  -f-  .69  is  the  same  as  that  of  412.62  -4-  69.  Here  we 
multiplied  each  number  by  100. 

5.08 

69)412.62 
345 
676 
621 
652 
552 

.-.  4.1262 --  .69  =  5.98. 

In  this  division  the  412  is  412  units,  and  the  quotient  5  is  therefore  5  units  ; 
676  is  676  tenths,  and  the  quotient  9  is  therefore  9  tenths ;  552  is  652  hun- 
dredths, and  the  quotient  8  is  therefore  8  hundredths. 

(2)  Find  correct  to  the  third  decimal  place  the  quotient 

of  8.94  -^  3.1416. 

3.1416)8.94 

2.845 

31416)89400 
62832 


265680 
251328 
143520 
125664 


178560 

157080 

21480 

.'.  8.94  -i-  3.1416  =  2.845,  correct  to  three  decimal  places. 

Exercise  117 
Divide,  proving  your  answer  correct  to  every  third  question 

1.  25.68-- 3.21.  3.    8.54^.07. 

2.  10.836  H- 5.16.  4.   $49.92^.065. 

5.  246.48 -.003;  61.725 -.075. 

6.  $64.26-^102;  $5100-5-1.02. 


DIVISION  OF  DECIMALS  189 

7.  $  5306.04  -5- 102  twice  in  succession. 

8.  $  84.3648  -;-  1.04  twice  in  succession. 

9.  $  54.75  -i-  .98,  correct  to  three  decimal  places. 
$  75.60  -T-  .99,  correct  to  three  decimal  places. 

10.  $  16989.7728  -j- 1.04  twice  in  succession  and  the  result  by 
1.02. 

11.  2450.90  -V-  .998,  correct  to  three  decimal  places. 

12.  $11,679-^4.8665. 

13.  .00081  -  27,  and  1.77089  by  4.735. 

14.  l-j-.l;  l-.Ol;  l-.OOOl. 

15.  31.5 -f- .126;  5.2 -.32. 

16.  12.6  ^  .0012,  and  .065341  -  .000475. 

17.  3.012  -  .0006. 

18.  130.4  -  .0004  and  .004 ;  46.634205  -  4807.65. 

19.  1.69  -  1.3,  by  .13,  by  13,  and  by  .013. 

20.  816^.0004. 

21.  .00005  --  2.5,  by  25,  and  by  .0000025. 

22.  32.5  -7-  8.7 ;  .02  —  1.7,  correct  to  four  decimal  places. 

23.  .009384  —  .0063,  correct  to  four  decimal  places. 

24.  37.24  —  2.9 ;  .0719  —  27.53,  correct  to  four  decimal  places. 

25.  Reduce  to  rods  :  38.5  yd. ;  10.45  yd. ;  4.4  yd. ;  7.315  yd. 

26.  Reduce  to  rods :  3.3  ft.;  .495  ft.;  379.5  ft.;  .33  yd. 

Exercise  118 

1.  A  conductor  whose  runs  amount  to  4000  mi.  a  month,  gets 
$  90  a  month.     This  is  how  many  cents  a  mile  ? 

2.  During  the  month  of  April,  1900,  the  United  States  exported 
to  Europe  cotton  to  the  value  of  $  24,684,078,  the  average  price 
being  9.3^  a  pound.     Find  the  number  of  pounds. 


190  ARITHMETIC 

3.  Find  the  cost  of  7225  lb.  coal  at  $  7.25  per  ton  of  2000  lb. 

4.  A  creditor  receives  $  1.50  for  every  $4  of  what  was  due  to 
him,  and  thereby  loses  $  301.05.     What  was  the  sum  due  ? 

5.  Divide  .0075  by  6.4,  and  explain  the  reason  for  fixing  the 
position  of  the  decimal  point  in  the  quotient. 

6.  A  merchant  expended  $280.60  in  purchasing  cloth  at  95^ 
a  yard,  at  f  1.37  a  yard,  and  at  73  ^  a  yard,  buying  the  same 
quantity  of  each.     Find  the  entire  number  of  yards  purchased. 

7.  Find  the  earth's  equatorial  diameter  in  miles,  supposing  the 
sun's  diameter,  which  is  111.454  times  as  great  as  the  equatorial 
diameter  of  the  earth,  to  be  883,345  mi. 

REDUCTION   OF  DECIMALS 
127.    (1)  Reduce  .275  to  a  common  fraction. 

.275  =  ^,\%  =  ^%%  =  H. 

(2)  Reduce  to  a  common  fraction  .08^. 

*  ~  100  ~  300  ~  12 

Exercise  119 
Reduce  to  common  fractions  in  their  lowest  terms: 


1. 

.5. 

6. 

.625. 

11. 

.334. 

16. 

8.9375. 

2. 

.25. 

7. 

.125. 

12. 

.03i. 

17. 

29.975. 

3. 

.75. 

8. 

.0625. 

13. 

.06^. 

18. 

18.06J. 

4. 

.60. 

9. 

.875. 

14. 

.66|. 

19. 

6.00|. 

5. 

.375. 

10. 

.16f. 

15. 

.14f 

20. 

249.075. 

21.  My  gain  on  selling  a  pound  of  tea  was  .125  of  the  cost, 
which  was  72  ^.     What  was  my  gain  on  1  lb.  ? 

22.  A  crate  of  berries  which  cost  $  1.35  was  sold  at  a  gain  of 
,22  J  of  the  cost  j  find  the  gain. 


REDUCTION  OF   DPXIMALS  191 

23.  I  bought  a  farm  for  $  4800,  and  sold  it  at  a  loss  of  .375  of 
the  cost  price.     Find  the  selling  price. 

24.  A  merchant  sold  coffee  at  a  gain  of  .33  J  of  the  cost.     His 
gain  on  a  quantity  of  coffee  was  $  12 ;  what  did  it  cost  him  ? 

25.  A  grain  merchant  sold  w^heat  for  ^  3400,  gaining  .06^  of 
the  cost.     Find  the  cost  price. 

128.    (1)  Divide  7  by  8,  expressing  the  result  as  a  decimal.- 

8)7.000 

.875 

Now  7 -8  =  |.     .-.  I  =.875. 
Proof 

(2)  Reduce  ^|  to  a  decimal. 

.9375 


16)150 
144 
60 
48 
120 
112 
80 
/.  If  =  .9375.  80 

(3)  Reduce  9||  to  a  decimal  correct  to  four  decimal  places. 

.7826 

2sJm 

161 
190 
184 
60 
46 
140 
138 
2 

.     .-.  9||  =  9.7826,  correct  to  four  decimal  places. 


192  ARITHMETIC 

Exercise  120 
Keduce  to  decimals  : 

1.  h  h  i-        2.  h  h  h  h        3.  j\,  ^,  A-        4.  ^%,  ii. 

5-    t\%  l¥jr,  ih'  6.    6i|,  3|f.  7.    f,  f,  e,  U- 

8.  Find  correct  to  four  decimal  places  the  value  of  ^,  |f, 

9.  Butter  bought  for  25^  a  pound  was  sold  for  28^  a  pound. 
Express  the  gain  as  a  decimal  of  the  cost. 

10.  I  bought  a  store  for  $6912,  and  sold  it  for  $  5184.     Ex- 
press the  selling  price  as  a  decimal  of  the  cost. 

11.  A  real  estate  agent  sold  land  which  cost  him  f  6240  at  a 
loss  of  $  2340.     What  was  his  loss  on  each  $  1000  invested  ? 

12.  A  chain  contains  66  ft.,  and  a  mile  5280  ft.     ^Vhat  decimal 
part  of  a  mile  is  a  chain  ? 


CHAPTER   XIII 

COMPOUND    QUANTITIES 

129.  Quantities  like  4  yd.,  S^  lb.,  and  6^  gal.  are  called 
simple  quantities,  because  they  are  expressed  in  terms  of  a 
single  unit  of  measurement. 

Quantities  like  3  lb.  8  oz.  6  gal.  1  qt.  are  called  compound 
quantities,  because  they  are  expressed  in  terms  of  two  or  more 
units  of  measurement. 

130.  The  units  of  money  are  the  units  which  are  used  to 
measure  the  values  of  things.  The  one  dollar  gold  piece  is  at 
present  (November,  1901)  the  prime  unit  or  standard  of 
value  in  the  United  States  and  Canada. 

131.  UNITS  OF  VALUE 
United  States  Money 

10  mills  (m.)  =  1  cent  (ct.  or  f) 
10  cents  =  1  dime  (d.) 

10  dimes         =  1  dollar  (|) 
10  dollars       =  1  eagle  (E.) 

The  coins  of  the  United  States  are  : 

Bronze  :  the  cent. 

Nickel :  the  five-cent  piece. 

Silver :  the  dime,  quarter-dollar,  half-dollar,  and  dollar. 

Gold  :  the  quarter-eagle,  half-eagle,  eagle,  and  double  eagle. 

132.  Sterling  Money  is  the  money  of  Great  Britain  and 
Ireland. 

o  193 


194  ARITHMETIC 

The  prime  unit  is  1  pound,  whose  value  is  $4.8665. 
The  pound,  when  coined,  is  called  the  sovereign. 

British  or  Sterling  Money 

4  farthings  (far. )  =  1  penny  (d. ) 
12  pence  =  1  shilling  (s.) 

20  shillings  =  1  pound  (£) 

5  shillings  =  1  crown 

21  shillings  =  1  guinea 

133.   The  unit  of  French  Money  is  1  franc,  which  is  worth 

19.8^. 

The  unit  of  German  Money  is  1  mark,  which  is  worth 

23.85^. 

Exercise  121 

1.  Howmany  mills  are  there  in  2^?   3^?   |^?   1^^?   2^^? 

2.  How  many  cents  are  there  in  40  mills  ?     60  mills  ?     15 
mills?     5  mills?     25  mills  ? 

3.  State  orally  the  table  of  English  Money. 

4.  Keduce  to  farthings  :  3d. ;  6d.  2  far. ;  9d.  3  far. 

5.  Eeduce  to  pence;  4s. ;  8s.  5d. ;  12s.  6d. 

6.  Reduce  to  shillings  :  £  7  ;  £2  12s. ;  £9  7s. 

7.  How  many  pence  are  there  in  ^s.  ?    Js.  ?    |s.  ?    |s.  ?    -|s.  ? 

8.  How  many  shillings  and  pence  are  there  in  £|?     d£^? 
£1?     £1? 

9.  How  many  shillings  are  there  in   £.3?     £.7?     £.25? 
£.33^ 

10.  What  fraction  of  a   shilling   is  3d.?    4c?.?    Sd.?    9d.? 
lOd.? 

11.  What  is  the  value  of  £  1  in  American  money  ?    Of  £  10  ? 
Of  £  100  ? 

12.  How  many  shillings  and  pence  are  there  in  60d.  ?    84d.  ? 
39cZ.?    58d.?    112d.? 


UNITS  OF  WEIGHT  195 

13.  How  many  pounds  and  shillings  are  there  in  80s.?   65s.? 
120s.?   48s.? 

14.  How  many  pounds  and  shillings  in  1  guinea  ?   4  guineas  ? 
6  guineas?    9  guineas? 

15.  What  decimal  of  a  pound  is  10s.  ?   12s.?    17s.?   24s.? 

16.  What  part  of  a  crown  is  Is.  ?     How  many  crowns  in  10s.  ? 
20s.? 

17.  Express  2  guineas  in  sovereigns  and  shillings. 

18.  What  is  the  value  of  10  francs  in  United  States  money  ? 

19.  What  is  the  value  of  100  marks  in  United  States  money  ? 

20.  What  is  the  cost  in  cents  of  3  books  at  1  franc  each  ? 

21.  What  is  the  difference  in  value  between  100  marks  and 
100  francs  ? 

UNITS  OF  WEIGHT 

134.   Avoirdupois  Weight  is  used  for  weighing  everything  ex- 
cept jewels,  precious  metals,  and  medicines  when  dispensed. 
The  prime  unit  of  weight  is  1  pound  Avoirdupois.     , 

Avoirdupois  Weight 

16  ounces  (oz.)       =  1  pound  (lb.) 
100  pounds  =  1  hundredweight  (cwt.) 

20  hundredweight  =  1  ton  (T.) 

In  the  United  States  Custom  House,  and  in  weighing  iron  and  coal  at  the 
mines,  the  long  hundredweight  and  the  long  ton  are  used. 

112  pounds  =  1  long  hundredweight 

2240  pounds  =  1  long  ton 

One  pound  Avoirdupois  =  7000  grains 
One  ounce  Avoirdupois  =  437 1  grains 

Exercise  122 

1.  State  orally  the  table  of  Avoirdupois  Weight. 

2.  Eeduce  to  ounces  :  1  lb.  8  oz. :  2  lb.  4  oz. 


196  ARITHMETIC 

3.  Express  1  lb.  8  oz.  as  a  fraction  of  2  lb.  4  oz. 

4.  Express  1  lb.  8  oz.  as  a  decimal  of  2  lb.  4  oz. 

5.  What  part  of  1  lb.  is  4  oz.  ?   12  oz.  ?   2  oz.?   8  oz.  ?     Ex- 
press your  results  also  as  decimals. 

6.  What  part  of  1  T.  is  400  lb.?   800  lb.  ?   1500  lb.? 

7.  A  coal  dealer  buys  coal  by  the  carload  at  the  mines.     How 
many  more  pounds  of  coal  does  he  get  for  1  T.  than  he  gives  ? 

8.  Show  by  dividing  7000  gr.  by  16,  that  1  oz.  Avoirdupois  is 
equal  to  437^  gr. 

9.  One  ounce  is  what  part  of  1  lb.  ?    ^  of  an  ounce  is  what 
part  of  lib.? 

10.   A  farmer  sells  3  cows  whose  united  weight  is  1  T.  5  cwt. 
What  is  the  average  weight  of  the  cows  ? 

135.   Troy  Weight  is  chiefly  used  for  weighing  gold,  silver, 

and  jewels. 

Troy  Weight 

24  grains  (gr.)      =  1  pennyweight  (pwt.) 
20  pennyweights  =  1  ounce  (oz.) 
12  ounces  =  1  pound  (lb.) 

One  pound  Troy  =  6760  grains 
One  ounce  Troy  =    480  grains 

Exercise  123 

1.  State  orally  the  table  of  Troy  Weight. 

2.  Reduce  to  grains:  1  pwt.  16  gr.;  2  pwt.  12  gr. 

3.  What  is  the  ratio  of  2  pwt.  12  gr.  to  1  pwt.  16  gr.? 

4.  Express  1  pwt.  16  gr.  as  a  decimal  of  2  pwt.  12  gr. 

5.  Reduce  1  oz.  to  grains. 

6.  Divide  5760  gr.  by  12,  and  verify  your  result  in  question  5. 


UNITS  OF  WEIGHT  197 

7.  By  how  many  grains  is  1  lb.  Avoirdupois  heavier  than  1  lb. 
Troy? 

8.  By  how  many  grains  is  1  oz.  Troy  heavier  than   1  oz. 
Avoirdupois  ? 

9.  How  many  ounces  and  pennyweights  are  there  in  J  lb.? 
fib.?   I  lb.?   fib.? 

10.  How  many  pennyweights  are  there  in  .25  oz.  ?  .4  oz.  ? 
.35  oz.? 

11.  What  part  of  a  pound  is  8  oz. ?    1  oz.?   |oz. ?   |oz.? 

12.  If  coal  is  worth  $6  a  ton, how  many  pounds  can  be  bought 
for  $3?   $2? 

13.  A  cubic  foot  of  water  contains  1000  oz.  How  many 
pounds  does  a  cubic  foot  of  water  weigh  ? 

136.  Druggists  buy  their  medicines  by  Avoirdupois 
Weight,  but  use  Apothecaries'  Weight  in  mixing  and  in 
selling  medicines. 

Apothecaries'  Weight 

20  grains  =  1  scruple  (3) 

3  scruples  =  1  dram  (3) 

8  drams  =  1  ounce  (  ^  ) 

12  ounces  =  1  pound  (lb) 

One  pound  Apothecaries'  Weight  =  5760  grains 
One  ounce  Apothecaries'  Weight  =   480  grains 

Exercise  124 

1.  State  orally  the  table  of  Apothecaries'  Weight. 

2.  How  many  ounces  in  16  3  ?   40  3?   72  3? 

3.  How  many  drams  in  9  3    18  3  ?   54  3  ? 

4.  What  part  of  a  pound  is45?   95?    10 5? 

5.  How  many  scruples  in  33  and  2^?     In  40  gr. ?     In  120 
gr.? 


198  ARITHMETIC 

UNITS  OF  LENGTH 

137.  The  prime  or  standard  unit  of  length  is  1  yard. 

Long  Measure 

12  Inches  (in.)  =  1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 

5^  yards  or  16 J  feet  =  1  rod  (rd.) 

320  rods  =  1  mile  (mi.) 

1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft. 

A  hand,  used  in  measuring  the  height  of  horses,  =  4  in,  ;  a  knot,  used  in 
navigation,  =  6086  ft.  or  1.15  mi. 

A  fathom,  used  in  measuring  depth  at  sea,  =  6  ft. 

138.  Surveyor's  Linear  Measure  is   used   by  surveyors   in 

measuring  land.     The  prime  unit  is  1  chain,  called  Gunters 

Chain, 

Surveyor's  Linear  Measure 

100  links  (1.)  =  1  chain  (ch.) 
80  chains       =  1  mile  (mi.) 

1  ch.  =  4  rd.  =  22  yd.  =  66  ft.  =  792  in. 
1  link  =7.92. 

139.  Mark  off  in  the  schoolroom  1  ft.,  1  yd.,  and  1  rd. 
Locate  two  points  exactly  1  mi.  apart. 

Exercise  125 

1.  How  many  inches  are   there  in   1   yd.?     J  yd.?    j  yd.? 

iyd.?    I  yd.? 

2.  Keduce  to  yards:  4  rd.;  8  rd. ;  32  rd.;  320  rd.;  1  mi. 

3.  Reduce  to  feet :  18  yd.;  76  yd.;  176  yd.;  1760  yd.;  1  mi. 

4.  Show  that  1  mi.  =  320  rd.  =  1760  yd.  =  5280  ft.  =  63,360  in. 

5.  How  many  yards  are  there  in  .3  mi.?     .8  mi.?     .25  mi.? 
M\  mi.? 

6.  What  part  of  a  foot  is  3  in.  ?     6  in.  ?     8  in.  ?     10  in.  ? 

7.  What  part  of  a  yard  is  12  in.  ?     18  in.  ?     24  in.  ?     27  in.  ? 


UNITS  OF   SURFACE   OR   SQUARE   MEASURE         199 

8.  What  part  of  a  rod  is  1  yd.  ?     3  yd.  ?     5  yd.  ?     5i  yd.  ? 

9.  What  part  of  a  mile  is  40  rd.  ?     200  yd.  ?     280  yd.  ? 

10.  How  many  yards  are  there  in  1  rd.  ?  How  many  feet  ? 
How  man}^  inches  ? 

11.  How  many  chains  are  there  in  320  rd.?  How  many  rods 
are  there  in  1  ch.  ? 

12.  Show  that  1  ch.  =  4  rd.  =  22  yd.  =  66  ft.  =  792  in. 

13.  How  many  inches  are  there  in  100  links  ?     In  1  link  ? 

UNITS   OF   SURFACE   OR   SQUARE   MEASURE 

140.  Surface  has  two  dimensions,  —  length  and  breadth. 

141.  The  prime  unit  of  area  is  1  square  yard,  which,  like 
1  square  inch,  1  square  foot,  1  square  rod,  and  1  square  mile, 
is  derived  from  the  corresponding  unit  of  linear  measure. 

The  measure  of  1  ft.  is  12,  the  unit  being  1  in. 
The  measure  of  1  sq.  ft.  is  144,  the  unit  being  1  sq.  in. 
.-.  1  sq.  ft.  =  144  sq.  in. 

The  measure  of  1  yd.  is  3,  the  unit  being  1  ft. 
The  measure  of  1  sq.  yd.  is  9,  the  unit  being  1  sq.  ft. 
.-.  1  sq.  yd.  =9  sq.  ft. 

The  measure  of  1  rd.  is  5|,  the  unit  being  1  yd. 

The  measure  of  1  sq.  rd.  is  (5|  x  5i),  or  30|,  the  unit  behig  1  sq.  yd. 
.-.  1  sq.  rd.  =  30^  sq.  yd. 

Illustrate  the  above  by  drawing  1  sq.  ft.,  1  sq.  yd.,  and  1  sq.  rd.,  and 
dividing  each  into  the  next  lower  units  of  area. 

In  the  case  of  1  sq.  rd.  draw  according  to  the  scale  of  4  in.  to  1  rd. 

Surface  or  Square  Measure 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 
9  square  feet  =  1  square  yard  (sq.  yd.) 

30^  square  yards  =  1  square  rod  (sq.  rd.) 

160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

10  square  chains  =  1  acre  ;  1  acre  =  4840  square  yards 


200 


ARITHMETIC 


142.   A  township  is  6  mi.  square,  and  is  divided,  as  in  the 
accompanying  figure,  into  36  sections,  each  1  mi.  square. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

W.  ^ 
320  ACRES 

160  ACRES 

N.  ^OF  S.E.  M 
80  ACRES 

8.  W.  X  OF 

8.E.   H 

40  ACRES 

8.E.)<0F 

8.E.  H 
40  ACRES 

TOWNSHIP. 


SECTION. 


Locate  sections  8,  22,  and  36  in  the  drawing.  Draw  a 
township  on  a  scale  of  1  in.  to  1  mi.,  and  divide  it  into 
36  sections,  numbering  each  section.  Divide  one  of  these 
sections  into  4  square  farms  of  160  A.  each,  and  name  each 
farm  according  to  its  position  in  the  section.  Divide  a 
second  section  into  8  rectangular  farms  of  80  A.,  and  a  third 
into  16  square  farms  of  40  A.,  and  locate  each  farm  as  before. 


isq.yd.? 
4  sq.  rd.  ? 


Exercise  126 

1.  How  many  square  inches  are  there  in  2  sq.  ft.  ?  4  sq.  ft.  ? 
9  sq.  ft.  ? 

2.  How  many  square  feet  in  5  sq.  yd.  ?    i  sq.  yd.  ? 

3.  1  sq.  rd.  is  equal  to  how  many  square  yards  ? 
16  sq.  rd.  ?     160  sq.  rd.  ?     1  A.  ? 

4.  What  part  of  a  square  rod  is  1  sq.  yd.  ? 

5.  Keduce  to  square  rods :  484  sq.  yd. ;  151 J  sq.  yd. 

6.  What  part  of  an  acre  is  80  sq.  rd.  ?     120  sq.  rd.  ? 

7.  How  many  square  chains  are  there  in  160  sq.  rd.  ? 
equals  how  many  square  rods  ? 


1  sq.  ch. 


UNITS  OF   VOLUME  201 

8.  Eeduce  to  acres  :  5  sq.  mi. ;  8  sq.  mi. ;  1  township. 

9.  If  1  sq.  mi.  is  the  unit  of  area,  find  the  number  which 
expresses  the  measure  of  2560  A.     Of  4  townships. 

10.  Into  how  many  townships  can  a  county  be  divided  which 
contains  324  sq.  mi.  ? 

11.  What  is  the  area  of  a  square  6  ft.  in  length  ? 

12.  What  is  the  difference  in  area  between  two  figures,  one 
6  in.  sq.  and  the  other  6  sq.  in.  ? 

Illustrate  by  drawing. 

UNITS  OF  VOLUME 

143.  A  volume  has  three  dimensions  —  length,  breadth, 
and  thickness. 

144.  The  prime  unit  of  volume  is  1  cubic  yard,  which, 
like  1  cubic  inch  and  1  cubic  foot,  is  derived  from  the  cor- 
responding unit  of  linear  measure. 

The  measure  of  the  volume  of  1  cu.  ft.  =  12  x  12  x  12  =  1728,  the  unit  of 
volume  being  1  cu.  in. 

The  measure  of  the  volume  of  1  cu.  yd.  =  3  x  3  x  3  =  27,  the  unit  of 
volume  being  1  cu.  ft. 

Cubic  or  Volume  Measure 

1728  cubic  inches  (cu.  in.)  =1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 

145.  Firewood  and  rough  stone  are  measured  by  the  cord 
(cd.).  The  cord  is  a  pile  8  ft.  long,  4  ft.  wide,  and  4  ft. 
high.  It  contains  128  cu.  ft.  One  cord  foot  (cd.  ft.)  is 
1  ft.  in  length  of  the  cord.     Its  volume  is  16  cu.  ft. 

A  cubic  yard  of  earth  is  called  a  load. 
How  many  loads  of  dirt  are  there  in  a  pile  15  ft.  long, 
12  ft.  wide,  and  6  ft.  deep  ? 

146.  Mark  off  in  one  corner  1  cu.  ft.,  1  cu.  yd.,  and  1  cd. 
Divide  the  cord  into  cord  feet. 


202  ARITHMETIC 

UNITS  OF  CAPACITY 

147.  The  prime  unit  of  capacity  is  1  gallon. 

Liquid  Measure 

4  gills  (gi.)  =  1  pint  (pt.) 
2  pints         =  1  quart  (qt.) 
4  quarts       =  1  gallon  (gal. ) 

148.  The  capacity  of  cisterns,  reservoirs,  and  the  like  is 
often  expressed  in  barrels  (bbl.)  of  31-J  gal.  each,  or  in  hogs- 
heads (hhd.)  of  63  gal.  each.  A  gallon  contains  231  cu.  in. 
Have  a  tin  box  made  11  in.  long,  7  in.  wide,  and  3  in.  deep, 
and  note  that  1  gal.  of  water  will  just  fill  it. 

Dry  Measure 

2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 

4  pecks  =  1  bushel  (bu.) 

One  bushel  contains  2150.42  cubic  inches 

149.  Apothecaries'  Fluid  Measure  is  used  by  druggists  in 
mixing  medicines. 

Apothecaries'  Fluid  Measure 

60  minims  (n\^)  =  1  fluid  dram  (f  3) 

8  fluid  drams    =  1  fluid  ounce  (f  5  ) 
16  fluid  ounces   =  1  pint  (0.) 

8  pints  =  1  gallon  (Cong.) 

One  minim  is  about  equal  to  1  drop 

Exercise  127 

1.  What  part  of  1  gal.  is  1  qt.  ?     1  pt.  ?     2  qt.  1  pt.  ? 

2.  What  is  the  number  of  cubic  inches  in  1  gal.  ?  In  1  qt., 
liquid  measure  ?     In  1  pt.,  liquid  measure  ? 

3.  What  part  of  1  bu.  is  1  qt.  ?     1  pt.  ?     3  pk.  4  qt.  ? 

4.  What  is  the  number  of  cubic  inches  in  1  bu.  ?  In  1  qt., 
dry  measure?     In  1  pt.,  dry  measure? 


UNITS  OF  TIME  203 

5.  How  many  more  cubic  inches  are  contained  in  1  qt.,  dry 
measure,  than  in  1  qt.,  liquid  measure  ? 

6.  Show  that  1  bu.  is  nearly  equal  to  9.31  gal. 

7.  Find  the  number  of  cubic  feet  in  1  cd. 

8.  Find  the  number  of  cords  of  wood  in  a  pile  30  ft.  long,  6  -ft. 
high,  and  4  ft.  wide. 

9.  Find  the  cost  of  a  pile  of  wood  24  ft.  long,  51  ft.  high,  and 
4  ft.  wide,  at  $  6  a  cord. 

UNITS  OF   TIME 
150.   The  prime  unit  of  time  is  1  day. 

Measure   of   Time 

60  seconds  (sec.)=  1  minute  (min.) 

60  minutes  =  1  hour  (hr.) 

24  hours  =  1  day  (da. ) 

7  days  =  1  week  (wk.) 

365  days  =  1  common  year  (yr.) 

366  days  =  1  leap  year  (1.  yr.) 
100  years  =  1  century  (C.) 

The  year  is  divided  into  12  calendar  months  : 

January  (Jan.) 31  da.  July 31  da. 

February  (Feb.)     .     .     28  or  29  da.  August  (Aug.) 31  da. 

March 31  da.  September  (Sept.)  .     .     .     .  30  da. 

April 30  da.  October  (Oct.) 31  da. 

May 31  da.  November  (Nov.)    ....  30  da. 

June 30  da.  December  (Dec.)    .     .     .     .  31  da. 

In  business  transactions,  1  mo.  is  generally  taken  as  equal  to  30  da.,  and 
1  yr.  as  equal  to  360  da. 

The  following  lines  are  useful  in  enabling  one  to  remember  the  number  of 
days  in  a  mouth  : 

"  Thirty  days  hath  September, 
April,  June,  and  November." 

A  year  is  the  period  of  the  earth's  revolution  about  the  sun.  It  consists 
of  365  da.  5  hr.  48  min.  50  sec. 

A  common  year  lacks  11  min.  10  sec.  of  being  365  da.  6  hr.,  or  365^  da. 
Hence,  when  we  take  365  da.  to  a  common  year  and  366  da.  to  a  leap  year. 


204  ARITHMETIC 

we  increase  each  year  by  11  min.  10  sec.  In  400  yr.  this  amounts  to  a  little 
over  3  da.  For  that  reason  three  out  of  four  centennial  years  are  counted  as 
common  years ;  i.  e.  the  centennial  years  that  do  not  divide  equally  by  400 
have  only  3(55  da. 

Exercise  128 

1.  Express  9  hr.  as  a  fraction  of  a  week. 

2.  Express  12  sec.  as  a  decimal  of  a  minute. 

3.  Express  146  da.  as  a  fraction  of  a  year. 

4.  Express  as  a  fraction  of  a  month :  10  da.,  15  da.,  18  da. 

5.  Express  as  a  decimal  of  a  month :  21  da.,  18  da.,  27  da. 

6.  Find  the  number  of  days  between  Jan.  3  and  Feb.  4 ;  March 
27  and  April  30 ;  Oct.  24  and  Dec.  11. 

7.  How  many  days  are  there  in  November?     January?    De- 
cember ?     April  ?     February  ? 

8.  What  part  of  a  year  each  is :  1  mo.  10  da.  ?    2  mo.  12  da.  ? 
7  mo.  6  da.  ?     (1  mo.  =  30  da. ;  1  yr.  =  360  da.) 

Exercise  129 

In  the  questions  in  the  following  exercise  find  the  day  on 
which  the  note  is  due.* 

Find  the  date  on  which  a  note  falls  due,  which  I  promise  to  pay : 

1.  Three  months  after  March  3,  1900. 

2.  Four  months  after  June  13,  1901. 

3.  Ninety  days  after  May  13,  1900. 

4.  Sixty  days  after  Sept.  16,  1899. 

5.  Ninety  days  after  June  4,  1901. 

Find  the  exact  number  of  days  between  the  day  on  which  each 
of  the  following  notes  is  discounted  and  the  day  on  which  it  is  due : 

6.  Day  of  discount,  May  7,  1901 ;  due  June  6,  1901. 

7.  Day  of  discount,  June  27,  1901 ;  due  Oct.  16,  1901. 

*  Add  three  days  (called  days  of  grace)  to  the  given  time  if  that  is  the 
custom  in  your  state. 


CIRCULAR  OR  ANGULAR  MEASURE 


205 


8.  Day  of  discount,  Sept.  4,  1901 ;  due  Oct.  30,  1901. 

9.  Day  of  discount,  Dec.  23,  1901 ;  due  Feb.  20,  1902. 
10.    Day  of  discount,  Jan.  15,  1901 ;  due  May  1,  1901. 


CIRCULAR  OR  ANGULAR  MEASURE 

151.  Angular  Measure  is  used  to  measure  arcs,  angles,  and 
in  determining  latitude,  longitude,  direction,  the  position  of 
vessels  at  sea,  and  the  like. 

152.  A  Circle  is  a  plane  figure  contained  by  one  line  called 
the  circumference,  all  points  of  which  are  equally  distant  from 
a  point  within  it  called  the  centre. 

One-half  of  the  circumference  is  called  the  semicircumfer- 
ence  and  one-fourth  a  quadrant. 

An  arc  is  any  portion  of  the  circumference. 

A  line  drawn  through  the  centre  and  terminated  at  both 
extremities  by  the  circumference  is  called  the  diameter. 

The  line  drawn  from  the 
centre  and  terminated  by 
the  circumference  is  called 
the  radius. 

In  the  figure  the  line  OB  has 
revolved  from  OA  through  one- 
fourth  of  a  revolution.  The  an- 
gle AOB  is  called  a  right  angle, 
and  contains  90°. 

OA  and  OB.  are  said  to  be  per- 
pendicular to  each  other. 

Angular  Measure 

60  seconds  (")=  1  minute  (') 
60  minutes        =  1  degree  (°) 
360  degrees         =  1  circumference  (C.) 

The  circumference  of  the  earth  at  the  equator  =  24,902  mi. 

The  length  of  a  degree  at  the  equator  =  24,902  mi.  -r-  360  =  69.17  mi. 


206  ARITHMETIC 

Exercise  130 

1.  What  part  of  a  revolution  is  1  right  angle  ?    2  right  angles? 
4  right  angles  ? 

2.  What  part  of  a  revolution  is  60°  ?     45°  ?     225°  ? 

3.  What  is  the  length  of  the  arc  of  1°  in  a  circle  whose  cir- 
cumference is  360  yd.  ? 

4.  If  an  arc  of  3°  is  6  ft.  long,  what  is  the  length  of  the  cir- 
cumference of  the  circle  ? 

MISCELLANEOUS  UNITS 

153.  Numbers  Paper 

12  units  =  1  dozen  (doz.)  24  sheets    =  1  quire 

12  dozen  =  1  gross  20  quires    =  1  ream 

12  gross  =  1  great  gross  2  reams     =  1  bundle 
20  units  =  1  score  5  bundles  =  1  bale 

MISCELLANEOUS  WEIGHTS 

154.  A  bushel  of  wheat  =  60  lb. 
A  bushel  of  beans  =  CO  lb. 
A  bushel  of  clover  seed  =  00  lb. 
A  bushel  of  shelled  corn  =  56  lb. 
A  bushel  of  rye  =  56  lb. 
A  bushel  of  barley  =  48  lb. 
A  bushel  of  oats  =  32  lb. 
A  bushel  of  potatoes  =  60  lb. 

A  bushel  of  coarse  salt  (domestic)  =  56  lb. 

These  are  the  legal  number  of  pounds  per  bushel  in  Michigan,  Indiana, 

Illinois,  Wisconsin,  Iowa,  Missouri,  and  New  York. 

On  the  Chicago  Board  of  Trade  seeds  are  sold  by  the  cental. 

A  barrel  of  flour  =  196  lb. 

A  barrel  of  pork  or  beef  =  200  lb. 
A  cental  of  grain  =  100  lb. 

Exercise  131 

1.   Pens  are  sold  in  boxes  containing  1  gross.    How  many  pens 
are  there  in  a  box  ? 


MISCELLANEOUS  WEIGHTS  207 

2.  Lead  pencils  are  sold  in  boxes  containing  i  gross.  How 
many  pencils  are  there  in  a  box  ? 

3.  How  many  packages  of  lead  pencils  of  1  doz.  each  are  there 
in  a  box  ? 

4.  Eggs  are  packed  in  crates  holding  30  doz.  How  many  eggs 
in  a  crate  ? 

5.  How  many  sheets  of  paper  in  20  quires  ? 

6.  What  articles  of  food  weigh  60  lb.  to  the  bushel  ? 

7.  How  many  bushels  of  wheat  weigh  as  much  as  10  bu.  of 
barley? 

8.  What  is  the  ratio  of  the  weight  of  1  bu.  of  barley  to  that 
of  1  bu.  of  oats  ? 

155.  A  certain  room  is  8  yd.  long.  Here  the  unit  of 
measurement  is  1  yd.,  and  the  measure  of  the  length  of 
the  room  is  the  number  8. 

The  area  of  the  floor  of  a  room  is  192  sq.  yd.  6  sq.  ft. 
Here  the  measure  of  the  area  is  the  sum  of  192  units  of 
1  sq.  yd.  and  6  units  of  1  sq.  ft. 

A  pitcher  holds  f  of  a  gallon  of  water. 

Here  the  measure  of  the  capacity  of  the  pitcher  is  |  and 
the  unit  of  measurement  is  1  gal. 

Exercise  132 

Name  the  measures  of  the  following  quantities,  and  the  units 
of  measurement : 

1.  The  volume  of  a  cistern  which  holds  450  cu.  ft. 

2.  The  volume  of  a  cistern  which  holds  1200  gal. 

3.  The  area  of  a  field  which  contains  S^  A. 

4.  The  value  of  a  house  worth  $  2800.  What  are  the  meas- 
ures of  the  value  of  the  house  with  the  following  units :  $  5,  $  10, 
f  50,  $  100  ? 


208  ARITHMETIC 

5.  The  weight  of  200  lb.  of  sugar.  What  are  the  measures  of 
the  weight  with  10  lb.,  1  cwt.,  and  1  T.  as  units  ? 

6.  The  weight  of  a  chest  of  tea  containing  65  lb.  If  5  lb.  is 
used  as  the  unit,  what  is  the  measure  ? 

7.  The  weight  of  8  oz.  of  gold.  If  1  pwt.  is  the  unit  of  meas- 
urement, what  number  expresses  the  measure?  What  number 
measures  the  weight  when  1  lb.  is  the  unit  ? 

8.  The  weight  of  40  gr.  of  quinine.  What  is  the  measure 
when  1  3  is  the  unit  ? 

9.  What  are  the  measures  of  1  lb.  of  gold,  of  lead,  and  of 
quinine,  when  the  unit  of  measurement  is  1  oz.  ?  When  1  gr.  is 
the  unit  ? 

10.  The  length,  of  the  circumference  of  a  circle  found  to  be 
22  yd.  long.  What  are  the  measures  of  the  circumference  when 
1  ft.  and  1  rd.  are  the  measures  ? 

11.  The  length  between  two  points  which  measures  4  ch. 
What  are  the  measures  of  the  length  when  the  units  are  1  mi. 
and  1  link  ? 

12.  The  area  of  a  field  which  contains  80  sq.  rd.  What  are 
the  measures  of  the  field  when  the  units  are  1  A.  and  1  sq.  yd.  ? 

13.  The  capacity  of  a  pitcher  which  contains  f  of  a  gallon  of 
water.     What  is  the  measure  when  the  unit  is  1  qt.  ? 

14.  The  capacity  of  a  basket  which  holds  6 J  qt.  What  are 
the  measures,  the  units  being  1  pt.  and  1  pk.  ? 

15.  The  time  of  a  rainstorm,  which  lasted  2  hr.  What  are  the 
measures,  the  units  being  1  min.  and  1  da.  ? 

16.  The  weight  of  a  silver  cup  which  is  15  oz.  12  pwt.  12  gr. 

156.  The  fundamental  units  used  in  the  measurement  of 
value  are  1  cent,  1  dime,  1  dollar,  and  1  eagle. 


MISCELLANEOUS  WEIGHTS  209 

The  value  of  a  postage-stamp  used  to  mail  a  letter  to  any 
part  of  the  United  States  or  Canada  is  measured  by  the 
number  2  and  the  unit  1^. 

The  cost  of  a  quart  of  berries  worth  15^  is  measured  by 
the  number  3  and  the  unit  1  nickel,  or  by  the  number  1  and 
the  unit  1  dime  plus  the  number  1  and  the  unit  the  nickel. 

It  may  also  be  measured  by  the  number  15  and  the  unit 
1^.  It  may  also  be  measured  by  the  number  1  and  the 
unit  1  quarter-dollar  less  the  number  1  and  the  unit  1  dime. 

The  unit  for  measuring  oil  is  1  gal.  That  for  buying  spice 
by  retail  is  1  oz.  Frequently  a  quantity  is  expressed  with 
reference  to  two  or  more  units.  Tims,  the  length  of  a  table 
being  1  yd.  2  ft.  6  in.,  the  units  are  1  yd.,  1  ft.,  and  1  in. 

Exercise  133 

1.  Name  instances  in  which  ^1  is  the  unit  of  value;  1^; 
1  nickel ;  1  dime.     What  unit  of  value  is  most  commonly  used  ? 

2.  Name  instances  in  which  the  units  of  weight  used  are  1  oz.; 
1  lb.;  1  cwt.;  1  T. 

3.  Name  quantities  whose  weight  is  measured  by  these  units : 
1  gr. ;  1  pwt. ;  1  oz. ;  and  1  lb. 

4.  Name  quantities  which  are  measured  by  the  unit  1  gr.; 
l3;  13;  15;  IR). 

5.  What  quantities  are  expressed  in  terms  of  these  units: 
1  in.?    1  ft.?   1yd.?   1  rd.?   1  mi.? 

6.  Name  things  whose  measurement  is  given  in  terms  of  the 
unit  1  pt. ;  1  qt. ;  1  gal. 

7.  Name  articles  whose  measurement  is  expressed  in  terms  of 
the  unit  1  bu. ;  1  pk. ;  1  qt. 

8.  In  measuring  time  give  instances  in  which  you  use  as  the 
unit  1  century;  1  yr.;  1  mo.;  1  wk.;  1  da.;  1  hr.;  1  min.;  1  sec. 


210  ARITHMETIC 

9.   Name  articles  whose  quantity  is  expressed  in  terms  of  the 
unit  1  doz. ;  1  gross ;  1  great  gross ;  1  score. 

10.  In  measuring  paper  the  following  units  are  used :  1  quire ; 
1  ream.  Give  instances  when  1  quire  is  used  as  the  unit,  and  also 
when  1  ream  is  used. 

11.  What  unit  of  weight  connects  Avoirdupois,  Troy,  and 
Apothecaries'  weight  ?     What  number  expresses  the  measure  of 

1  lb.  of  each  kind  in  terms  of  the  common  unit  ?     Of  1  oz.  ? 

-12.    What  units  are  common  to  Apothecaries'  and  Troy  weight 
and  of  equal  value  ? 

13.  What  units  of  length  connect  Surveyors'  Long  Measure 
with  Linear  Measure  ? 

14.  Name  five  units  of  area  which  are  derived  from  corre- 
sponding units  of  length.  Why  is  1  A.  chosen  as  a  unit  of  area? 
Give  instances  in  which  1  A.  is  used  as  the  unit  of  area,  and  also 
when  1  sq.  mi.  is  the  unit. 

15.  Name  a  unit  of  volume  larger  than  1  cu.  yd.  Why  have 
we  no  units  of  volume  corresponding  to  the  linear  units  1  rd. 
and  1  mi.  ? 

16.  Find  the  number  of  cubic  inches  in  1  qt.,  dry  measure,  and 
find  how  much  greater  it  is  than  1  qt.,  liquid  measure. 

17.  State  the  number  of  days  in  the  years  1600,  1600,  1700, 
1800,  1900,  2000. 

18.  The  circumference  of  a  circle  is.  1,296,000  in.  in  length. 
Find  the  length  of  1°;  of  1';  of  1". 

19.  Find  the  quantity  measured  by  the  number  4  and  the  unit 

2  ft.  6  in.     By  the  number  8  and  the  unit  2  gal.  3  qt. 

20.  How  many  square  rods  in  1  A.  ?  How  many  acres  in 
1  sq.  mi.  ?  What  part  of  a  square  mile  is  a  farm  of  40  A.  ?  If 
this  farm  is  in  the  form  of  a  square,  what  is  the  length  of  one 
side  ?     Of  the  perimeter  ? 


REDUCTION  211 


REDUCTION 

X57,  Reduction  Descending  is  the  process  of  reducing  a 
quantity  expressed  in  terms  of  a  unit  or  units  of  measure- 
ment to  a  quantity  expressed  in  terms  of  a  smaller  unit,  or 
of  smaller  units  of  measurement. 

Reduce      2  mi.  36  rd.  5  yd.  2  ft.  to  feet. 
320 
640 
36 
676  rd. 

_M 

338 

3380 

g  by  the  law  of  commutation  as  320  x  2  rd.,  and 

gi^    jj  not  as  representing  320  x  2  mi. 


Note.  —  In  this  reduction  we  are  to  think 
of  the  operation  as  signifying  2  x  320  rd.,  or 


3 


11169 
2 


11171  ft. 

2  mi.  =  2  X  320  rd.  =  640  rd. 

2  mi.  36  rd.  =  640  rd.  +  36  rd.  =  676  rd. 

676  rd.  =  676  x  5|  yd.  =  3718  yd. 

2  mi.  36  rd.  5  yd.  =  3718  yd.  +  5  yd.  r^  3723  yd. 

3723  yd.  z=  3723  x  3  ft.  =  11,169  ft. 

/.  2  mi.  36  rd.  5  yd.  2  ft.  =  11,169  ft.  +  2  ft.  =  11,171  ft. 

Exercise  134 

1.  Reduce  5  gal.  3  qt.  1  pt.  to  pints. 

2.  Reduce  18  bu.  6  pk.  3  qt.  to  quarts. 

3.  Reduce  £5  12  s.  9d.  to  pence. 

4.  Reduce  8  oz.  15  pwt.  17  gr.  to  grains. 

5.  Reduce  7  T.  18  cwt.  14  lb.  to  pounds. 

6.  Reduce  15  da.  17  min.  to  minutes. 

7.  Reduce  18  rd.  4  yd.  2  ft.  6  in.  to  inches. 

8.  Reduce  5  oz.  5  dr.  2  sc.  16  gr.  to  grains. 


212  ARITHMETIC 

9.    Reduce  2  A.  4  sq.  rd.  2  sq.  yd.  8  sq.  ft.  to  square  feet. 

10.  Reduce  16  cu.  ft.  1374  cu.  in.  to  cubic  inches. 

11.  Eeduce  1  mi.  18  rd.  2  yd.  2  ft.  6  in.  to  inches. 

12.  State  how  to  reduce  a  quantity  from  higher  to  lower  de- 
nominations. 

13.  Find  the  number  of  acres  in  a  township. 

14.  Find  the  number  of  cubic  inches  in  a  vessel  containing 
25  gal. 

15.  Reduce  1  lb.  7  oz.  14  pwt.  19  gr.  to  grains. 

16.  Reduce  7  T.  15  cwt.  56  lb.  to  pounds. 

17.  Reduce  17  lb.  2  §  2  9  to  grains. 

18.  Reduce  3  cu.  yd.  1001  cu.  in.  to  cubic  inches. 

19.  Reduce  760  bu.  3  pk.  to  quarts. 

20.  Reduce  6  reams  19  quires  18  sheets  to  sheets. 

21.  Reduce  5  da.  17  hr.  to  seconds ;  and  1  wk.  23  hr.  59  sec.  to 
seconds. 

22.  Reduce  5  A.  80  sq.  rd.  28  sq.  yd.  to  square  yards. 

158.  Reduction  Ascending  is  the  process  of  reducing  a 
quantity  expressed  in  terms  of  a  unit,  or  of  units,  to  a  quan- 
tity expressed  in  terms  of  a  larger  unit,  or  of  larger  units. 

(1)  Reduce  242,337  in.  to  higher  denominations. 

242337 


11 

320 


20194  ft.  9  in. 


6731  yd.  1  ft. 
2 


13462  half  yd. 


1223  rd.  9  half  yd.,  i.e.  4  yd.  1  ft.  6  in. 


3  mi.  263  rd. 
.*.  242,337  in.  =  3  mi.  263  rd.  4  yd.  1  ft.  9  in. 


1  tt.  y  m.  -» 
1  ft.  6  in.  J 
=  3  mi.  263  rd.  5  yd.  0  ft.  3  in. 


KEDUCTION  213 

242,337  in.  =  20,194  ft.  9  in. 
20,194  ft.  9  in.  =  6731  yd.  1  ft.  9  in. 
6731  yd.  1  ft.  9  in.  =  1223  rd.  ^  yd.  1  ft.  9  in. 
1223  rd.  4i  yd.  1  ft.  9  in.  =  3  mi.  263  rd.  4^  yd.  1  ft.  9  in. 
=  3  mi.  263  rd.  5  yd.  0  ft.  3  in. 
.'.  242,337  in.  =  3  mi.  263  rd.  5  yd.  3  in. 

(2)  To  prove  this  answer  correct,  reduce  3  mi.  263  rd.  5  yd.  3  in.  to 
inches,  by  the  method  of  the  preceding  exercise. 

159.  We  reduce  a  quantity  expressed  in  terms  of  a  smaller  unit  to  larger 
units  in  order  to  get  a  more  definite  idea  of  its  value. 

Thus  we  form  no  definite  idea  of  a  distance  between  two  points  when  we 
are  told  that  it  is  242,337  in. ;  but  we  have  a  definite  idea  of  the  same  dis- 
tance when  we  are  told  that  it  is  3  mi.  263  rd.  5  yd.  3  in. 

Exercise  135 

Reduce  to  higher  denominations,  and  prove  every  third  answer 
correct : 

1.  678  pt.  4.   4728  cu.  ft.  to  cords. 

2.  4622  pt.  of  dry  measure.      5.   18,420  lb.  of  wheat  to  bushels. 

3.  483,197  sec.  6.   21,489d 

7.  93,742  oz. 

8.  State  how  to  reduce  a  quantity  expressed  in  terms  of  a 
lower  unit  to  higher  units. 

9.  5420  gr. 

10.  141,728  gr.  14.  364,428  in. 

11.  57,893  cu.  in.  15.  273,460  sq.  yd. 

12.  56,735d  16.  6,188,724  sq.  in. 

13.  38,297  oz.  .  17.  429,678  in. 

18.  73,940  K. 

19.  89,673  gr.,  Apothecaries'  weight. 

20.  7493  units. 

21.  Reduce  37,921  in.  to  chains,  rods,  etc. 

22.  Reduce  121,838  A.  to  townships,  etc. 


214  ARITHMETIC 

COMPOUND  ADDITION   AND  SUBTRACTION 
160.   Add: 


mi. 

rd. 

yd. 

ft. 

in. 

2 

27 

1 

2 

8 

1    • 

146 

2 

1 

6 

8 

91 

2 

0 

4 

7 

152 

1 

2 

9 

19 

97 

^ 

1 

3 

i 

=  1 

6 

19  97        2  2        9 

The  sum  of  the  hiches  colutnn  is  27  in.,  or  2  ft.  3  in. 

The  sum  of  the  feet  column,  increased  by  2  ft.,  is  7  ft.,  or  2  yd.  1  ft. 

The  sum  of  the  yards  column,  increased  by  2  yd.,  is  8  yd.,  or  1  rd.  2\  yd. 

The  sum  of  the  rods  column,  increased  by  1  rd.,  is  417  rd.,  or  1  mi.  97  rd. 

The  sum  of  the  miles  column,  increased  by  1  mi.,  is  19  mi. 

Changing  \  yd.  to  1  ft.  6  in.  and  adding,  we  have  the  sum  =  19  mi.  97  rd. 
2  yd.  2  ft.  9  in. 

As  in  the  problems  in  addition  in  Chapter  IV,  we  are  required  in  the  pre- 
ceding question  to  find  the  whole  quantity  measured  by  the  four  given  parts, 
of  which  the  first  is  2  mi.  27  rd.  1  yd.  2  ft.  8  in.  What  are  the  other  three 
measured  parts  ? 

In  this  question,  how  would  the  work  of  writing  and  adding  be  diminished 
if  our  units  of  length  were  arranged  according  to  the  decimal  system  ? 

161.   Subtract  53  lb.  5  oz.  18  pwt.  from  72  lb.  4  oz.  7  pwt. 


lb. 

oz. 

pwt. 

72 

4 

7 

53 

5 

18 

18        10  9 

Since  we  cannot  take  18  pwt.  from  7  pwt.,  take  1  oz,  or  20  pwt.  from  4  oz. 
and  add  it  to  the  7  pwt.,  making  27  pwt.  18  pwt.  from  27  pwt.  leaves  9  pwt. 
Since  we  cannot  take  5  oz.  from  3  oz.,  take  1  lb.  or  12  oz.  from  72  lb.  and 
add  it  to  the  3  oz.,  making  15  oz.  6  oz.  from  15  oz.  leaves  10  oz.  63  lb.  from 
71  lb.  leaves  18  lb.     Hence  the  difference  =  18  lb.  10  oz.  9  pwt. 

As  in  the  problems  in  subtraction  in  Chapter  V,  we  are  given  in  the  pre- 
ceding question  the  whole  quantity  measured  by  72  lb.  4  oz.  7  pwt.,  and  one 
part,  viz.  53  lb.  5  oz.  18  pwt.,  and  are  required  to  find  the  part  measured  by 
their  difference. 


COMPOUND   ADDITION   AND   SUBTRACTION  215 


Add: 

Exercise  136 

bu. 

pk. 

qt.     pt. 

T. 

cwt. 

lb. 

1.  3 

5 

6    1 

4. 

16 

17 

74 

8 

4 

1    0 

13 

10 

20 

7 

3 

5    1 

17 

15 

19 

9 

4 

3    1 

84 
11 

0 
11 

87 
36 

lb. 

2.  18 
16 
23 
17 

oz. 

11 

9 
8 
6 

pwt. 

16 

22 

6 

13 

5. 

3 

22 

56 

3 
15 

3 

3 
0 

2 
6 

3 

2 
1 

2 

1 

19 

10 

11 

9 

£ 

s. 

d. 

79 

4 

1 

10 

3.   5 

17 

10 

cu.  yd. 

cu.  ft. 

cu.  in. 

36 

0 

11 

6. 

3 

23 

171 

7 

3 

4 

17 

17 

31 

73 

19 

8 

28 

26 

1000 

30 

14 

5 

34 

23 

1101 

7.  State  how  to  add  compound  quantities. 

Subtract: 

lb. 

oz. 

pwt. 

ft) 

3 

3 

3 

8.  144 

8 

14 

10. 

144 

9 

4 

1 

106 

11 

16 

129 

0 

7 

_3 

y<i- 

ft. 

in. 

lb. 

oz. 

pwt. 

gr. 

9.  15 

1 

5 

11. 

5836 

0 

0 

0 

13 

2 

7 

cu.  vd. 

12.  37 

35 

cu.  ft 

18 
24 

4976 

cu.  in. 

857 
1280 

7 

13 

19 

13.  State  how  to  subtract  oae  compound  quantity  from  another. 

14.  State  in  what  respects  addition  and  subtraction  of  com- 
pound quantities  are  the  same  as  addition  and  subtraction  of 
numbers,  and  state  how  they  differ. 


216  ARITHMETIC 

COMPOUND  MULTIPLICATION  AND  DIVISION 

162.   Multiply  5  wk.  6  da.  18  hr.  by  11. 

We  are  here  required  to  find  the  whole  quantity  measured  by  the  uuit 
6  wk.  6  da.  18  hr.  and  the  number  11. 


wk. 

da. 

hr. 

5 

6 

18 
11 

65        4  6 

We  multiply  18  hr.  by  11  and  obtain  the  product  198  hr.,  or  8  da.  6  hr. 
Then  we  multiply  6  da.  by  11  and  obtain  the  product  66  da.,  which,  increased 
by  8  da.,  is  74  da.,  or  10  wk.  4  da.  Multiplying  5  wk.  by  11,  and  adding 
10  wk.,  we  have  65  wk.     Hence  the  product  is  65  wk.  6  da.  18  hr. 

What  is  the  ratio  of  65  wk.  4  da.  6  hr.  to  5  wk.  6  da.  18  hr.  ? 

163.    (1)  Divide  88  rd.  3  yd.  1  ft.  by  34. 


rd.    yd. 

ft. 

rd. 

y<i. 

ft. 

34)  88    3 

1 

(2 

3 

1 

68 

20  rd. 

5i 

110  yd. 

3 

118  yd. 

102 

11yd. 

3 

33  ft. 

1 

34  ft. 

34 

Hence  the  quotient  or  unit  of  measure  is  2  rd.  3  yd.  1  ft. 

The  remainder  on  dividing  88  rd.  by  34  is  20  rd.  20  rd.  3  yd.  =  113  yd. 
The  remainder  on  dividing  113  yd.  by  34  is  11  yd.     11  yd.  1  ft.  =  34  ft. 

On  dividing  34  ft.  by  34,  there  is  no  remainder.  Hence  the  quotient  is 
2  rd.  3  yd.  1  ft.    What  part  of  the  dividend  is  the  quotient  ? 


COMPOUND  MULTIPLICATION  AND  DIVISION       217 

(2)  Divide  73  gal.  1  pt.  by  16  gaL  1  qt. 

We  are  here  required  to  find  the  number  which  is  the  ratio  of  the  quan- 
tity 73  gal.  1  pt.  to  the  unit  16  gal.  1  qt. 

73  gal.  1  pt.  =  585  pt.  ;  16  gal.  1  qt.  =  130  pt. 

585  pt.  -4-  130  pt.  =  4^. 

.*.  the  quotient  or  ratio  is  4|. 

Exercise  137 

1.  Multiply  7  gal.  3  qt.  1  pt.  by  9. 

2.  Multiply  8  da.  12  hr.  25  min.  by  8. 

3.  Divide  £199  6s.  Sd.  by  13. 

4.  Divide  459  lb.  4  oz.  5  pwt.  22  gr.  by  29. 

5.  Multiply  86  lb.  7  oz.  16  pwt.  11  gr.  by  8. 

6.  Multiply  5  wk.  6  da.  18  hr.  14  min.  by  11. 

7.  Divide  1738  cu.  yd.  382  cu.  in*  by  302. 

8.  Divide  684  da.  8  hr.  9  min.  by  47. 

9.  Multiply  70  yd.  2  ft.  10  in.  by  7. 

10.  Divide  1  mi.  54  rd.  1  ft.  2  in.  by  29. 

11.  Multiply  2  hr.  8  min.  9  sec.  by  15. 

12.  Divide  13°  26' by  15. 

13.  (a)  State  how  to  multiply  a  compound  quantity  by  a  given 
number.  (6)  State  how  to  divide  a  compound  quantity  by  a 
given  number. 

14.  State  in  what  respects  multiplication  and  division  of  com- 
pound quantities  are  like  multiplication  and  division  of  numbers, 
and  how  they  differ. 

FRACTIONS  OF   SIMPLE   AND   COMPOUND   QUANTITIES 
164.    (1)  Find  the  value  of  f  of  a  mile. 

f  mi.  =  f  of  320  rd.  =  266f  rd. 
I  rd.  =  I  of  5i  yd.  =  3f  yd. 
I  yd.  =  I  of  3  ft.       =  2  ft. 
.-.  I  of  a  mile  =  266  rd.  3  yd.  2  ft. 


218  ARITHMETIC 

(2)  Find  the  value  of  f  bu.  —  f  pk. 

f  bu.  =  ^  of  4  pk.  =  2f  pk. 
2f  pk.  -  ^  pk.         =  le  pk. 
l§pk.  =  Hof8qt.  =  4^qt. 
.-.  I  bu.  -  f  pk.  =  1  pk.  i^  qt. 

Exercise  138 
Find  the  value  of : 

1.  |of  £1;  I  of  £2. 

2.  y\  of  a  day ;   j^  of  a  mile. 

3.  I  of  3  T. ;  7|  lb.  Avoir. 

4.  I  bu.  -  f  pk. ;  I  lb.  Troy  +  f  lb.  Troy  -  |  oz.  Troy. 

5.  I  A. ;  I  A.  -  I  sq.  rd. ;  |  sq.  mi. 

6.  I  of  5  lb.  8  oz.  6  pwt. 

7.  ^  of  2  mi.  38  rd.  4  yd.  2  ft.  2  in. 

8.  State  how  to  express  a  fraction  of  a  unit  of  measure  in 
terms  of  smaller  units. 

Express  .854  of  an  acre  in  lower  denominations. 
.854  A. 

160 


51240 

854 

136.640  sq. 

rd. 

30t 
16 

1920 

19.36  sq. 

yd. 

9 
8.24  sq. 

ft. 

144 
96 

96 

24 

34.56  sq. 

in. 

.-.  .854  A.  =  136  sq.  rd.  10  sq.  yd.  3  sq.  ft.  34.56  sq.  in. 
Adapt  note,  §  157,  to  this  problem. 


SIMPLE   AND   COMPOUND   QUANTITIES  219 

Exercise  139 

Find  the  value  of : 

1.  .84  of  a  day.  4.    5.923  mi.  -  75.18  rd. 

2.  .045  of  a  mile.  5.    £  7 5 A3  -  16.76s. 

3.  .6  of  a  pound  Troy.  6.    4.7  A.  -  2.93  sq.  rd. 

165.  Reduce  213  rd.  5  ft.  6  in.  to  the  fraction  of  3  mi. 

213  rd.  5  ft.  6  in.  =  42,240  in. 

3  mi.  =  3  X  63,360  in.  =  190,080. 

.♦.  213  rd.  5  ft.  6  in.  =  j%\%,  or  f  of  3  ml. 

The  G.  C.  M.  of  42,240  and  190,080  is  21,120,  which  divides  the  numerator 
twice  and  the  denominator  9  times. 

Exercise  140 

1.  Reduce  £1  7s.  6d.  to  the  fraction  of  £2. 

2.  Reduce  266  rd.  3  yd.  2  ft.  to  the  fraction  of  a  mile. 

3.  Reduce  14  hr.  24  min.  to  the  fraction  of  a  day. 

4.  Reduce  4  mo.  3  da.  to  the  fraction  of  a  year.     (30  da.  to  a 
month,  and  360  da.  to  a  year.) 

5.  Reduce  10  mo.  24  da.  to  the  fraction  of  a  year. 

6.  Reduce  1|-  in.  to  the"  fraction  of  1|-  yd. 

7.  Reduce  f  lb.  Avoirdupois  to  the  fraction  of  5  lb.  Troy. 

8.  Express  1  yd.  2  ft.  6  in.  as  a  fraction  of  2  yd.  1  ft.  6  in. 

166.  Express  58  rd.  2  yd.  7.2  in.  as  a  decimal  of  a  mile. 

7.2  in. 


12 

3 

5.5 

320 


.6  ft. 


2.2  yd. 


58.4  rd. 


.1825  mi. 
7.2  in.  =  .6  ft.  =  .2  yd. 
2  yd.  +  .2  yd.  =  2.2  yd.  =  .4  rd. 

58.4  rd.  =  .1825  mi. 
.-.  58  rd.  2  yd.  7.2  in.  =  .1825  mi. 
Prove  this  answer  correct  by  reducing  .1825  mi.  to  lower  denominations. 


220  ARITHMETIC 

Exercise  141 

1.  Eeduce  8  oz.  15.2  pwt.  to  the  decimal  of  a  pound. 

2.  Reduce  21  hr.  57  min.  36  sec.  to  the  decimal  of  a  day. 

3.  Reduce  147  rd.  1  yd.  3.6  in.  to  the  decimal  of  a  mile. 

4.  25  sq.  mi.  128  A.  is  what  decimal  of  a  township  ? 

5.  Reduce  67  sq.  rd.  6  sq.  yd.  64.8  sq.  in.  to  the  decimal  of 
an  acre. 

Board  Measure 

167.  Boards,  planks,  joists,  etc.,  are  sold  by  the  board  foot. 
The  board  foot  is  equal  in  volume  to  one  square  foot  of 
board,  one  inch  thick. 

Thus  a  board  18  ft.  long,  14  in.  wide,  and  1  in.  thick  or  less  contains 
18  X  ^1,  or  21  ft.,  board  measure. 

To  find  the  number  of  feet,  board  measure,  in  lumber  more 
than  1  inch  thick,  we  find  the  number  of  square  feet  in  the 
surface  of  the  board  and  multiply  this  result  by  the  number 
of  inches  that  the  lumber  is  thick. 

Thus  a  board  15  ft.  long,  8  in.  wide,  and  2^  in.  thick  contains  15  x  y^  x  |, 
or  25  board  feet. 

Exercise  142 

How  many  feet,  board  measure,  in : 

1.  A  board  20  ft.  long,  9  in.  wide,  and  1  in.  thick  ?   Jin.  thick? 

2.  A  board  18  ft.  long,  8  in.  wide,  and  2^  in.  thick  ? 

3.  A  scantling  16  ft.  long,  3  in.  wide,  and  4  in.  thick  ? 

4.  Twenty  scantlings,  24  ft.  long,  5  in.  wide,  and  7  in.  thick  ? 

5.  A  stick  of  timber  33  ft.  long  and  14  in.  square? 

6.  One  cubic  foot  ? 

7.  Find  the  number  of  board  feet  in  2  doz.  of  each  of  the 
following  scantlings : 

(1)  2  X  4  X  10  ft.        (4)  2  X  4  X  20  ft.        (7)  3  x  10  x  16  ft. 

(2)  2  X  6  X  14  ft.        (5)  2  X  8  X  18  ft.        (8)  4  x  6  x  18  ft. 

(3)  2  X  10  X  24  ft.       (6)  2  X  12  X  22  ft.       (9)  6  X  6  x  10  ft. 


LOis'GITUDE  AND  TIME  221 

8.  What  is  the  cost  of  25  joists  each  6  in.  by  4  in.  by  15  ft. 
at  f  22  per  thousand? 

9.  What  is  the  cost  of  24  joists  each  5  in.  by  7  in.  by  10  ft. 
at  $21  per  thousand?  / 

10.  How  much  will  it  cost  to  enclose  a  rectangular  lot  50  ft. 
wide  and  100  ft.  deep  with  a  tight  board  fence  6  ft.  high  with 
boards  that  cost  |18  per  thousand? 

11.  Find  the  cost  of  the  lumber  needed  for  a  tight  board  side- 
walk 100  ft.  long,  4  ft.  wide,  2  in.  thick,  if  sold  in  16  ft.  lengths 
at  $  16  per  thousand. 

Longitude  and  Time 

168.  Turn  to  your  geography  and  find  several  meridian 
lines.  Find  the  prime  meridian  which  passes  through  Green- 
wich, England. 

The  imaginary  lines  drawn  on  the  earth's  surface  from 
pole  to  pole  are  called  meridians.  The  meridian  passing 
through  Greenwich,  a  town  near  London,  England,  hav- 
ing the  royal  observatory,  is  called  the  prime  or  standard 
meridian. 

Places  west  of  the  prime  meridian  are  in  west  longitude, 
and  places  east  of  the  prime  meridian  are  in  east  longitude. 
Thus,  Washington  is  77°  1'  west  longitude,  and  Paris  2°  20' 
east  longitude.  _  __ 

169.  Find  from  the  maps  in  your  geographies  to  the 
nearest  degree  the  longitude  of  these  cities :  New  York, 
Pittsburg,  Richmond,  Atlanta,  Chicago,  Denver,  Salt  Lake 
City,  San  Francisco. 

Find  also  the  longitude  of  Rome,  Stockholm,  Athens, 
Constantinople,  St.  Petersburg,  and  Moscow. 

170.  The  difference  in  longitude  between  Philadelphia, 
which  is  75°  9'  west  longitude,  and  Portland,  which  is  70°  15' 
west  longitude,  is  4°  54'. 


222  ARITHMETIC 

The  difference  between  the  longitude  of  Philadelphia  and 
that  of  Paris,  which  is  2°  20'  east  longitude,  is  77°  29',  and 
is  obtained  by  finding  the  sum  of  the  longitudes. 

171.  Find  on  the  map  of  the  United  States  and  name  the 
meridians  that  denote  Eastern  time,  Central  time,  Mountain 
time,  and  Pacific  time.  How  many  degrees  are  there  be- 
tween these  meridian  lines  ?  What  is  the  difference  in  time 
between  places  situated  on  these  meridians  ? 

172.  As  the  sun  rises  in  the  east,  it  is  sunrise  in  New  York 
earlier  than  in  Chicago;  consequently  at  any  time  during 
the  day  the  clock  time  in  New  York  is  later  than  in  Chicago. 
Similarly,  clock  time  in  San  Francisco,  which  is  west  of 
Chicago,  is  earlier  than  in  the  latter  city. 

173.  Since  the  sun  appears  to  move  in  a  circle  about  the 
earth,  i.e.  through  360°  in  24  hr.,  we  have  the  following: 

In  24  hr.  the  sun  passes  through  360°. 

In    1  hr.  the  sun  passes  through  15°. 

In  1  min.  the  sun  passes  through  ^  of  15°  =  J°  =  15'. 

In  1  sec.    the  sun  passes  through  ^^  of  15'  =  |'  =  15". 

Hence.,  to  reduce  longitude  expressed  in  time  to  lorigitude 
expressed  in  degrees.,  we  multiply  hy  15,  and  to  reduce  longitude 
expressed  in  degrees  to  longitude  expressed  in  time.,  we  divide 
hyl5. 

174.  Make  the  multiplication  table  of  15  and  memorize  it,  so  as  to  be 
able  to  work  questions  in  longitude  and  time  by  short  multiplication  and 
division. 

Exercise  143 

1.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  1  hr.  ?  2  hr.  ?  4  hr.  ?  2  min.? 
3  min.  ?    1  sec.  ?    3  sec.  ? 


LONGITUDE  AND   TIME  223 

2.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  1  hr.  2  min.  ?   1  hr.  3  niin.  2  sec.  ? 

3.  What  is  the  difference  in  time  between  two  places  whose 
difference  in  longitude  is  30°  ?  75°?  120'?  90'?  135"?  105"? 

4.  What  is  the  difference  in  time  between  two  places  whose 
difference  in  longitude  is  15°  45'  30"?   75°  15'  45"  ? 

5.  Find  the  difference  in  longitude  between  the  following 
places,  and  illustrate  your  answers  by  diagrams : 

Washington  77°  west  longitude  and  Helena  112°  west  longitude. 
Washington  77°  west  longitude  and  Hamburg  10°  east  longitude. 
Cairo  32°  east  longitude  and  Hamburg  10°  east  longitude. 

6.  What  is  the  difference  in  time  between  two  places: 

(1)  One  64°  west  longitude,  the  other  34°  east  longitude  ? 

(2)  One  64°  west  longitude,  the  other  26°  east  longitude  ? 

(3)  One  64°  east  longitude,  the  other  34°  east  longitude  ? 

7.  When  it  is  6  a.m.  at  San  Francisco,  what  time  is  it  at 
a  place  45°  east  of  San  Francisco?  30°  east?  15° 45'  east? 
30°  15' 45"  east? 

8.  When  it  is  11  a.m.  at  Chicago,  what  time  is  it  at  a  place  60° 
west  of  Chicago ?   30°  west?   45' west?   15° 45'? 

175.  (1)  Find  the  difference  in  time  between  St.  Louis  90° 
19' 26"  west  longitude  and  Sacramento  121°  25' 41''  west 
longitude. 

121°     25'  41' 

90       19  26 

15)31°       6'  15"       difference  in  longitude 

2  hr.  4  min.  25  sec.  difference  in  time 

A  difference  in  longitude  of  31°  gives  a  difference  of  31  -f-  15,  or  2  hr.  of 
time,  with  a  remainder  1°.  A  difference  in  longitude  of  1°6',  or  66',  gives 
a  difference  of  66  -^  15,  or  4  min.  of  time,  with  a  remainder  6'.  A  difference 
of  6'  15",  or  375",  gives  a  difference  of  375  -f- 15,  or  25  sec.  of  time. 


224  ARITHMETIC 

(2)  Berlin  is  13°  23'  53'^  east  longitude  and  Boston  is  71° 
4' 9"  west  longitude.  When  it  is  1.15  p.m.  at  Boston,  what 
time  is  it  at  Berlin  ? 

13°       23'  53" 

71  4  9 

15)84°       28'  2"  difference  in  longitude 

5  hr.  37  min.  52 j^  sec.  difference  in  time 
1  hr.  15  min. time  in  Boston 

6  hr.  52  min.  52^^  sec.  time  in  Berlin 

.*.  it  is  52  min.  52^2^  sec.  after  6  p.m.,  or  7  min.  7{|  sec.  to  7  p.m. 

Exercise  144 
Find  the  difference  in  time  between  the  following  cities : 

1.  Brooklyn  73°  58'  W.  and  Omaha  95°  28'  W. 

2.  St.  Paul  93°  3' 45"  W.  and  Cleveland  81°  39'  W. 

3.  Indianapolis  86°  6' 57"  W.  and  San  Francisco  122°  26'  12"  W. 

4.  Cincinnati  84°28'36"  W.  and  Glasgow  4°17'6"  W. 

5.  Detroit  83° 5'  7"  W.  and  Vienna  16°  22'22"  E. 

6.  Pillsbury  79°  55' 43"  W.  and  Amsterdam  4° 52' 13"  E. 

7.  Newark  74°9'12"  W.  and  Rome  12°27'58"  E. 

8.  When  it  is  11  a.m.  at  Cleveland,  what  o'clock  is  it  at 
St.  Paul? 

9.  What  time  is  it  at  Indianapolis  at  the  opening  of  school  at 
9  A.M.  in  San  Francisco  ? 

10.  When  it  is  7  a.m.  at  Cincinnati,  what  time  is  it  at  Glasgow  ? 

11.  When  it  is  8  a.m.  at  Omaha,  what  time  is  it  at  Brooklyn? 

12.  A  man  travels  until  his  watch  is  1  hr.  5  min.  16  sec.  slow. 
Does  he  travel  east  or  west,  and  how  many  degrees  has  he  gone  ? 

13.  A  vessel  sailed  from  a  port  directly  on  a  line  of  latitude 
a  certain  distance,  and  then  due  north  to  port,  where  the  captain 
found  that  his  chronometer  was  40  min.  slow.  In  which  direction 
did  he  sail  at  first,  and  how  many  degrees  ? 


ROOFING,   PAPERING,   ETC.  225 

14.  What  is  the  difference  in  longitude  between  two  places 
whose  difference  in  time  is  : 

(a)  2  hr.  33  min.  18  sec.  ? 

(6)  4  hr.  27  min.  46  sec.  ? 

(c)  6  hr.  12  min.  29  sec.  ? 

15.  Buffalo  is  78°57'48"  W.  and  Constantinople  is  28°59'3"  E. 
What  time  is  it  in  Constantinople  when  it  is  20  min.  after  6  a.m., 
July  6,  in  Buffalo  ? 

16.  What  time  is  it  in  Buffalo  when  it  is  20  min.  after  6  a.m., 
July  6,  in  Constantinople  ? 

17.  Given  the  longitude  of  two  places,  state  how  to  find  the 
time  in  the  place  east  at  a  given  time  in  the  place  west. 

ROOFING,  PAPERING,   ETC. 

176.  A  square  of  roofing  is  100  sq.  ft. 

177.  Shingles  are  16  in.  long  and  are  estimated  to  average 
4  in.  in  width. 

178.  Wall  paper  is  sold  by  the  double  roll  16  yd.  long  and 
18  in.  wide,  or  by  the  single  roll  8  yd.  long  and  18  in.  wide. 

179.  Bricks  are  usually  8  in.  x  4  in.  x  2  in.  In  estimating 
the  amount  of  work  or  material,  masons  measure  the  length 
of  walls  on  the  outside,  thus  counting  the  corners  twice. 

Exercise  145 

1.  Find  the  areas  of  these  rectangles : 

24  ft.  by  16  ft.  40  ft.  6  in.  by  24  ft. 

18  ft.  by  12  ft.  32  ft.  8  in.  by  18  ft. 

2.  Find  the  cost  of  the  roofing  for  a  barn,  each  side  of  the  roof 
being  40  ft.  long  and  25  ft.  wide,  at  $  5.13  a  square. 


226  ARITILMKTIC 

3.  Find  the  cost  of  the  roofing  for  a  buihling,  each  side  being 
(1)  4G  ft.  8  in.  long  and  24  ft.  wide,  at  f  5  a  square. 

(2)  37  ft.  6  in.  long  and  26  ft.  8  in.  wide,  at  f  3  a  square. 

4.  How  many  shingles  that  average  4  in.  in  width  and  are  laid 

(1)  4  in.  to  the  weather  will  cover  a  square  ? 

(2)  4^  in.  to  the  weather  will  cover  a  square  ? 

(3)  6  in.  to  the  weather  will  cover  a  square  ? 

5.  How  many  thousand  shingles,  laid  4  in.  to  the  weather,  are 
needed  to  cover  both  sides  of  a  roof  60  ft.  long  and  25  ft.  wide  ? 
What  is  the  cost  at  $  3.30  a  thousand  ? 

6.  A  roof  is  covered  with  shingles  laid  4^  in.  to  the  weather. 
Find  their  cost  at  f  2.90  a  thousand,  the  roof  being  90  ft.  long 
and  30  ft.  wide. 

7.  How  many  yards  of  carpet,  J  yd.  wide,  will  be  required 
for  a  room  16  ft.  long  and  12  ft.  wide,  the  carpet  running  length- 
wise ?     Find  its  cost  at  75  ^  a  yard. 

8.  Find  the  cost  of  the  hemp  carpet,  f  yd.  wide,  required  for 
a  room  18  ft.  long  and  16  ft.  wide  at  18  ^  a  yard,  the  carpet  run- 
ning lengthwise. 

9.  Find  the  cost  of  the  carpet,  J  yd.  wide,  required  for  a  room 
24  ft.  long  and  20  ft.  wide,  the  carpet  running  lengthwise  and 
costing  $  .96  a  yard,  there  being  a  waste  in  matching  of  4^  in.  on 
each  breadth  except  the  first. 

10.  A  double  roll  of  wall  paper  is  16  yd.  long  and  18  in.  wide. 
How  many  square  feet  does  it  contain? 

11.  A  room  is  16  ft.  long,  12  ft.  wide,  and  9  ft.  high.  How 
many  double  rolls  of  wall  paper  are  required  to  paper  the  four 
walls  ?     Find  the  cost  at  23  ^  a  double  roll. 

12.  A  room  is  18  ft.  long,  15  ft.  wide,  and  10  ft.  high  and  con- 
tains two  windows  and  one  door.  How  many  double  rolls  of  wall 
paper  are  required  for  the  four  walls,  an  allowance  of  20  sq.  ft. 
being  made  for  each  opening  ? 


COMPOUND  QUANTITIES  227 

13.  How  many  bricks  8  in.  x  4  in.  x  2  in.  will  make  1  cu.  ft.  ? 
Note.  —  Including  mortar,  allow  22  bricks  for  1  cu.  ft.  of  wall. 

14.  What  is  the  volume  of  a  solid  wall  24  ft.  x  3  ft.  x  9  in.  ? 

15.  How  many  bricks  will  be  required  to  build  a  solid  wall 
36  ft.  long,  4  ft.  high,  and  2  bricks  thick? 

Exercise  146 

1.  Find  the  length  measured  by  the  number  4  and  the  unit 
1  ft.  6  in. 

2.  Find  values  of  the  quantity  measured  by  the  number  6 
according  as  the  unit  is  £  2  5s.  or  6  oz.  10  pwt.  16  gr. 

3.  Find  the  value  of  a  pile  of  cordwood  16'  long  by  6'  high  at 
^  4  a  cord. 

4.  Find  the  value  of  a  pile  of  cordwood  13' 4"  long  by  3' 9" 
high  at  $4.50  a  cord. 

5.  An  English  sovereign  weighs  123.274  gr.  How  many 
sovereigns  will  weigh  61637  gr.  ? 

6.  If  a  sovereign  weighs  123.274  gr.,  how  many  sovereigns 
will  weigh  21  lb.  4  oz.  16  pAvt.  10  gr.  ? 

7.  A  race  course  is  170  ft.  shorter  than  |  mi.  How  many 
feet  long  is  it  ? 

8.  A  railway  bridge  over  the  Des  Moines  Eiver  is  40  ft.  more 
than  i  mi.  long.     How  many  feet  long  is  it  ? 

9.  An  ocean  steamer  sailed  across  the  Atlantic,  a  distance  of 
2780  knots,  in  6  da.  2  hr.  Find  the  average  number  of  knots  per 
hour. 

10.  The  steamer  Kaiser  made  an  ocean  voyage  of  3044  knots 
in  5  da.  18  hr.     Find  the  average  rate  per  hour. 

11.  How  many  feet  in  1  mi.  ? 

A  railway  station  in  Peru  is  16,635  ft.  above  the  sea.     This  is 
how  many  yards  more  than  3  mi.  ? 


228  ARITHMETIC 

12.  How  many  square  yards  in  1  A.  ?     Square  feet  ? 

The  Boers  of  the  Transvaal  were  allotted  40,000  sq.  ft.  at  the 
Paris  Exposition.     This  is  how  many  square  feet  less  than  1  A.  ? 

13.  Wire  fencing  costs  6^  per  yard.;  find  what  must  be  paid 
for  enclosing  a  field  305  yd.  long  and  156  yd.  wide,  there  being 
4  rows  of  wire. 

14.  Convert  £6  16s.  sterling  into  dollars  and  cents,  £1  being 
worth  $4.8665. 

15.  Find  the  selling  price  of  one  bale  of  cotton  (1  bale  =  500  lb.) 
at  4|d.  a  pound.    Eeduce  the  result  to  pounds,  shillings,  and  pence. 

16.  A  box  3  ft.  long  and  2  ft.  wide  contains  12  cu.  ft.    Find 

its  depth. 

17.  A  ravine  400  ft.  long  and  80  ft.  deep  is  filled  by  dumping 
into  it  128,000  cu.  yd.  of  earth.     Find  its  width. 

18.  A  race  horse  has  a  record  of  1  min.  15  sec.  for  6  furlongs. 
At  the  same  rate  how  long  would  he  take  to  go  1  mi.  ? 

19.  A  bicyclist  rode  39  mi.  in  3  hr.  15  min.  Find  the  rate  in 
miles  per  hour. 

20.  Find  the  number  of  days,  hours,  and  minutes  in  -^  yr. 
+  yV  wk.  4- 1^  br. 

21.  Make  out  the  following  account  neatly,  accurately,  and  in 
proper  form.     All  fractions  are  to  be  retained. 

John  Wilson  bought  from  you  to-day : 

7^  lb.  cheese  @  12^  ^  per  pound ; 
6J  lb.  butter  @  23^  per  pound ; 
2J  lb.  tea       @  55  ^  per  pound ; 
27  lb.  sugar  @  |  1  per  18  lb. 

22.  A  yard  measure  is  ^  of  an  inch  too  long.  What  is  the 
actual  distance  between  two  points  which  is  found  by  this  measui-e 
to  be  288  yd.? 


COMPOUND  QUANTITIES  229 

23.  Express  1  lb.  Troy  as  the  fraction  of  1  lb.  Avoirdupois ; 
express  1  lb.  Avoirdupois  as  the  fraction  of  1  lb.  Troy. 

24.  Which  is  the  heavier,  a  pound  of  gold  or  a  pound  of 
feathers,  and  an  ounce  of  gold  or  an  ounce  of  feathers  ?  By  how 
much  in  each  case  ? 

25.  How  many  silver  spoons,  each  weighing  2  oz.  16  pwt., 
could  be  made  out  of  a  bar  of  silver,  the  weight  of  which  is  50  oz. 
8  pwt.  ? 

26.  A  man  bought  a  quantity  of  tea  supposed  to  be  done  up  in 
packages  of  1  lb.  each,  for  which  he  was  to  pay  $  64 ;  on  weigh- 
ing them,  however,  it  was  found  that  each  package  was  1  oz.  too 
light.     How  much  should  he  pay  for  the  tea  ? 

27.  A  balloon  passes  over  114  mi.  in  285  min.  Show  that  this 
is  at  the  rate  of  2  mi.  in  5  min. 

28.  What  part  of  an  hour  is  1  min.  15  sec?  A  horse  ran 
I  mi.  in  1  min.  15  sec.     Find  the  rate  in  miles  per  hour. 

29.  The  winner  of  the  automobile  race  at  the  Paris  Exposition 
made  351  mi.  in  19  hr.  9  min.     Find  the  rate  in  miles  per  hour. 

30.  In  a  100-mi.  bicycle  race  Charles  Andrews  was  given  a 
handicap  of  2  hr.  45  min.  over  E.  H.  Smith.  Charles  Andrews 
finished  the  distance  in  7  hr.  58  min.  30  sec.  and  E,.  H.  Smith  in 
5  hr.  57  min.  40  sec.     By  how  much  did  the  former  win  ? 

31.  Find  the  number  of  tons  of  steel  rails  required  for  a  rail- 
way track  170  mi.  long,  weighing  90  lb.  to  the  yard. 

32.  A  lot  150  ft.  long  and  100  ft.  wide  is  to  be  surrounded 
by  a  close  board  fence  6  ft.  high.  What  will  the  boards  cost  at 
$  12.50  per  thousand  feet  ? 

33.  If  a  room  is  12  ft.  square,  what  must  its  height  be  in  order 
that  the  area  of  the  walls  may  amount  to  60  sq.  yd.  ? 

34.  Find  the  value  of  a  rectangular  field  330  yd.  by  160  yd.  @ 
$  36.50  per  acre. 


230  ARITILMETIC 

35.  Find  the  volume  of  a  rectangular  block  3'  9"  x  2'  4"  x  V  3". 

36.  Express  as  a  fraction  of  an  acre  the  sum  of  the  following: 
1.  of  4  of  II  of  1  A. ;  I  of  ff  of  If  of  100  sq.  rd. ;  and  ^  of  2J 
times  600  sq.  yd. 

37.  The  Manufacturers  and  Liberal  Arts  Building  of  the  Colum- 
bian Fair  was  in  the  form  of  a  rectangle  and  covered  an  area  of 
30  A.  76  sq.  rd.  19  sq.  yd.  7  sq.  ft.  The  building  was  787  ft. 
wide.     How  many  feet  in  length  was  it  ? 

38.  A  200-acre  farm  is  sown  with  grain  as  follows:  Peas,  25  A. 
126  sq.  rd.  10  sq.  yd.;  oats,  46  A.  134  sq.  rd.  15  sq.  yd.;  wheat, 
75  A.  125  sq.  rd.  25  sq.  yd.  The  buildings,  garden,  and  orchard 
occupy  12  A.,  and  the  rest  is  pasture.  How  many  acres  of  pasture 
are  there  ? 

39.  If  a  road  is  4  rd.  wide,  how  many  miles  of  it  will  make 
10  A.  ? 

40.  A  map  is  drawn  to  a  scale  of  half  an  inch  to  a  mile.  How 
many  acres  are  represented  by  a  square  inch  on  the  map  ? 

41.  After  drawing  off  124  gal.  of  water  from  a  cistern,  ^  of 
the  water  still  remained.  How  many  gallons  did  the  cistern  at 
first  contain?     How  many  gallons  were  left  in  it? 

42.  Some  Atlantic  liners  consume  198  T.  of  coal  per  day. 
They  average  8  da.  out  and  8  back.  For  fear  of  accidents  they 
carry  a  supply  for  4  da.  extra.  How  many  cubic  yards  of  the 
hold  of  such  a  steamer  will  be  occupied  with  coal  for  her  round 
trip  if  each  ton  is  33  cu.  ft.  ? 

43.  A  pile  of  wood  12  ft.  long,  4  ft.  wide,  and  6  ft.  high  was 
sold  for  $  13.50. 

(a)  What  was  the  price  per  cord  ? 

(b)  At  $  4  per  cord,  what  would  the  load  be  worth  ? 

44.  Find  the  value  of  a  pile  of  tan  bark  180  ft.  long,  48  ft. 
wide,  and  16  ft.  high  at  $2.25  per  cord. 


COMPOUND  QUANTITIES  231 

45.  Find   the    amount    of    the   following  bill,   retaining   all 

fractions : 

3|lb.  tea  @80^; 

300  1b.  sugar  @4|^; 

45  yd.  print  @lli^; 

2^  gal.  syrup  @  65^; 

12-1- yd.  towelling  @  121^; 

I  doz.  knives  and  forks  @  $  2.50 ; 

27  1b.  cheese  @  15)^; 

1  lb.  10  oz.  lemon  peel  @  32  ^  per  pound. 

46.  Make  a  drawing  to  show  how  many  yards  a  train  80  yd. 
long  must  go  to  cross  a  bridge  140  yd.  long.  This  is  what  part 
of  1  mi.  ? 

47.  A  train  80  yd.  long  crossed  a  bridge  140  yd.  long  in  221  sec. 
Find  the  average  speed  of  the  train  while  crossing. 

48.  Find  the  weight  of  a  bar  3  yd.  1  ft.  9  in.  long,  of  which  a 
yard  weighs  15  lb. 

49.  Find  the  cost  price  of  lead  per  hundredweight,  if  the  sale 
of  48  cwt.  for  $218.70  gives  a  profit  of  ^  of  the  original  price. 

50.  Find  the  expense  of  fencing  a  railway  (both  sides)  73  mi. 
in  length,  at  the  rate  of  $5.50  per  rod. 

51.  If  a  wheel  makes  260  revolutions  in  passing  over  1170  yd., 
what  is  its  circumference  ? 

52.  A  block  of  stone  is  4  ft.  long,  2  ft.  6  in.  broad,  and  1  ft. 
3  in.  thick ;  it  weighs  27  cwt.  Find  the  weight  of  50  cu.  in.  of 
the  stone. 

53.  A  rectangular  lot  45  ft.  front  by  99  ft.  deep  was  sold  for 
$3150.  What  was  the  price  per  foot  frontage,  and  what  the 
price  per  acre,  at  the  rate  of  the  selling  price  of  the  lot  ? 

54.  If  I  buy  147  gallons  of  molasses  at  19^  a  gallon,  and  use 
33  gallons  of  it,  at  how  much  must  I  sell  the  remainder  per 
gallon  so  as  to  receive  as  much  as  the  whole  cost? 


232  ARITHMETIC 

55.  When  1  oz.  of  gold  costs  $  19.45,  what  is  the  cost  of  .04  lb.  ? 

56.  A  grocer  receives  $9.60  for  a  bill  of  goods  weighed  on 
scales  that  gave  only  15J  oz.  to  the  pound.  How  many  cents' 
worth  did  he  cheat  his  customer  ? 

57.  If  a  cow  gives  12  qt.  1  pt.  of  milk  every  day,  and  1  lb.  8  oz. 
of  butter  can  be  made  from  25  qt.  of  milk,  how  many  pounds  of 
butter  can  be  made  in  one  week  from  the  milk  of  16  cows  ? 

58.  Find  the  expense  of  sodding  a  plot  of  ground,  which  is 
40  yd.  long  and  100  ft.  wide,  with  sods  each  a  yard  in  length  and 
a  foot  in  breadth;  the  sods  when  laid  costing  75^  per  hundred. 

59.  Make  out  the  following  bill  neatly  and  accurately.  John 
Smith,  a  merchant  of  Chicago,  sold  to  William  Jones,  on  June 
15,  1895 : 

5  lb.  8  oz.  of  butter     @  16  ^  per  pound ; 
2  lb.  10  oz.  of  tea        @    3^  an  ounce ; 
4  doz.  lemons  @    4^  for  3  lemons; 

8  lb.  coffee    "  @  37^^  per  pound; 

I  bu.  3  pk.  chestnuts  @  10^  per  quart; 

II  doz.  penholders      @    1^^  each. 

60.  Find  cost  of  digging  a  cellar  48  ft.  long,  30  ft.  wide,  and 
6  ft.  deep,  at  20^  per  cubic  yard,  and  flooring  it  with  Portland 
cement  at  10^  per  square  yard. 

61.  How  many  bushels  of  potatoes  can  be  sold  out  of  a  garden 
in  which  there  are  160  rows  of  potatoes,  in  each  row  240  hills,  and 
on  an  average  10  potatoes  in  each  hill,  if  128  potatoes  make  1  bu.  ? 

62.  Farmer  B.  sold  to  a  merchant  the  following  articles  to 
apply  on  an  overdue  account  of  $54.45: 

1680    lb.  of  hay  @  $  15  per  ton ; 

3|  cd.  of  wood  @  $4.80  per  cord; 

4    bbl.  of  apples      @  $  2.75  per  barrel ; 
350    lb.  of  flour  @  $2.50  per  hundredweight  5 

30    lb.  10  oz.  butter  @  16^  per  pound. 


COMPOUND  QUANTITIES  233 

Make  out  the  account  neatly,  showing  the  balance  and  to  whom 
due. 

63.  The  times  when  game  may  be  shot  in  Illinois  are  :  wild 
turkeys,  from  Sept.  1  to  Jan.  15  inclusive ;  quail,  from  Nov.  1  to 
Dec.  20 ;  turtle  doves,  from  Sept.  1  to  Dec.  1 ;  water  fowl,  from 
Sept.  1  to  April  15. 

Find  the  number  of  days  each  year  for  which  the  above  game 
are  protected. 

64.  The  price  of  crushed  stone  was  advanced  from  $1.15  to 
f  1.50  a  cubic  yard.  Find  the  increase  in  the  cost  of  laying  a 
road-bed  of  crushed  stone  1  mi.  long,  12  ft.  wide,  and  6  in.  deep. 

65.  Name  the  quantities  that  are  measured  by  the  following 
numbers :  1760,  1728,  5280,  5760,  7000,  231,  7.48,  3.1416. 


CHAPTER  XIV 

PERCENTAGE 

180.  The  expressions  ^yoq  ^^^^  i.05  denote  that  the 
quantity  1 1  is  conceived  as  made  up  of  100  equal  parts  or 
units,  and  that  5  of  these  parts  or  units  have  been  taken  to 
measure  the  quantity  denoted  by  j^q  or  .05. 

The  phrase  per  cent  means  hundredths.     Thus  the  fraction 
j^-Q  and  the  decimal  .05  are  also  written  5  per  cent  or  5^. 
Hence  yf^,  .05,  and  5^  of  any  quantity  are  equal. 

181.  (1)  Express  J  as  hundredths  and  also  as  per  cent. 

4  =  .33i  or— ^  =  33i%. 
J  '100         ^ '°^ 

(2)  A  horse  dealer  who  had  600  horses  sold  480  of  them. 
What  per  cent  of  his  horses  did  he  sell  ? 

We  are  here  given  the  measured  quantity  or  480  horses  to  compare  with 
the  quantity  600  horses,  and  we  are  required  to  find  the  per  cent  which  is  the 
number. 

The  number  sold  =  |^g  or  f  or  80  %  of  the  whole  number. 

182.  The  term  per  cent  is  used  constantly  in  business. 
The  merchant  gains  20^6,  meaning  that  he  gains  120  on 
every  8100  he  has  invested  in  goods.  The  insurance  com- 
pany charges  2f)  for  insuring  furniture,  meaning  that  §2  is 
charged  on  every  $100  worth  of  furniture  insured.  A  man. 
borrows  money  at  5^,  meaning  that  he  is  to  pay  $5  interest 
on  every  $100  borrowed.  The  commission  merchant  charges 
2^  of  the  buying  or  selling  price.  The  broker  charges  ^Jfc 
for  buying  stocks,  and  so  on. 

234 


PERCENTAGE  235 

Exercise  147 
Express  as  hundredths  and  also  as  per  cent : 

1  1.1.3.4.1.2.4.5.     1.     3.     7.      1.3.     9.17.     1.11.23 
-■■•      2  J    45   45    45    5J  "J)    55    55    10  J    10)    11^5    2TJ5    2^)    2  "5  5    2"0"5  ^"5  5    2'5'5   2"5' 

2  1  •    -2  •    i  •     5  .     1.    3.-     5.    i  .    JL  •      1    .      1    .    15..     17.     14.     41 
'^'      35     35     65     65     85     85     85     85     125     l3^J     1^)     165T95     2T5     4?' 

3.  Out  of  a  class  of  25  pupils  5  are  absent.  What  part  of  the 
class  is  absent  ?  How  many  hundredths  ?  What  per  cent  of 
the  class? 

4.  A  merchant  paid  $3  for  hats  which  he  sold  for  $4.  What 
fraction  of  the  cost  price  did  he  gain?  How  many  hundredths? 
What  per  cent  of  the  cost  ? 

5.  A  person  bought  a  house  for  ^5000  and  afterward  sold  it 
for  $4000.  The  loss  was  what  fraction  of  the  cost?  How  many 
hundredths  ?     What  per  cent  ? 

6.  A  fruit  dealer  bought  strawberries  for  $1.75  a  crate  and 
sold  them  for  $  2.25  a  crate.     What  per  cent  did  he  gain  ? 

7.  How  do  you  find  the  gain  per  cent  when  you  are  given  the 
cost  price  and  the  selling  price  of  an  article  ? 

8.  A  man  bought  a  horse  for  $234  and  afterward  sold  it  for 
$273.     What  per  cent  of  the  cost  did  he  gain? 

9.  The  population  of  a  town  of  32,000  inhabitants  increases 
1120  in  one  year.     What  is  the  per  cent  of  increase  ? 

183.   Express  25  fo  and  37 1^  as  fractions  in  their  lowest 

^^°~100     200     8 

Exercise  148 

What  fractions  in  their  lowest  terms  are  equivalent  to  the 
following : 

1.  1%;4%;  5%;  10%;  20%;  25%;  30%;  40%;  80%;  90%; 
100%? 

2.  60%;  75%;  35%;  24%;  70%;  16%; 


236  ARITHMETIC 

3.  12i%;  37i%;  621%;  87J%? 

4.  6i%;  8-1%;  6|%;  16|%  ;  83i%  ? 

5.  1H%;  142%;9,V%? 

6.  10f%;  5|%;  1||%  ? 

7.  100%  ;  120%  ;  125% ;  175%  ;  250%  ;  325%  ? 

8.  A  horse  which  cost  $  120  was  sold  at  a  gain  of  25%.  The 
gain  is  equal  to  what  part  of  the  cost  ?  How  much  was  gained  ? 
What  was  the  selling  price  ? 

9.  Cloth  which  cost  60^  per  yard  was  sold  at  a  loss  of  16|%. 
The  loss  was  what  fraction  of  the  cost  ?  What  was  the  loss  on 
each  yard  ?     What  was  the  selling  price  per  yard  ? 

10.  An  article  costing  $  4.20  was  sold  at  a  gain  of  8J%.  Find 
the  gain.     Find  the  selling  price. 

11.  How  do  you  find  the  gain  on  an  article  when  you  are  given 
the  cost  price  and  the  gain  per  cent  ?  How  do  you  hud  the  selling 
price  ? 

12.  Tea  is  bought  for  84^  per  pound  and  sold  at  an  advance  of 
14^%.     What  was  the  selling  price  of  each  pound? 

13.  A  drover  sold  400  sheep  at  a  gain  of  10%.  He  gained  the 
cost  price  of  how  many  sheep  ? 

184.  The  following  results  should  be  memorized  so  that 
the  fractions  or  the  per  cent  can  be  given  rapidly  in  any 
order : 


20%  =i 

33i%=i 

37i%=f 

100%  =  1 

40%  =1 

66|%=f 

50%  =  i 

6i%=TV 

60%  =  f 

100%  =  1 

62i%=f 

6|%=tV 

80%=^ 

12i%  =  i 

75%=} 

84%=  A 

100%  =  1 

25%  =i 

87i%  =  J 

16§%  =  J 

Note.  — The 

expression  20  %  = 

J  signifies  that  20%  of 

a  quantity  =  ^  of 

PERCENTAGE  237 

Exercise  149 
Kead  the  following  decimals  as  per  cents : 

1.  .25;  .16|;  .40;  .75;  .03^;  1.20;  1.25;  2.50;  3.16. 

2.  .15;  .371;  .451 ;  .051;  .OOJ;  2.40;  .001;  .06^;  .06|. 

185.   I  sold  an  article  that  cost  1 840  at  a  gain  of  16|  ^. 
Find  the  gain. 

The  gain  =  16|  %  or  ^  of  $  840  =  $  140. 

Exercise  150 

1.  I  sold  an  article  that  cost  $  720  at  a  gain  of  8^%.  What 
was  the  gain  ?     The  selling  price  ? 

2.  I  sold  an  article  that  cost  $465  at  a  loss  of  20%.  What 
was  the  loss  ?     The  selling  price  ? 

3.  I  sold  an  article  that  cost  $  885  at  a  gain  of  6|%.  What 
was  the  gain  ?     The  selling  price  ? 

4.  I  sold  an  article  that  cost  $  1275  at  a  loss  of  32%.  What 
was  the  loss  ?     The  selling  price  ? 

5.  How  do  you  find  the  gain  when  given  the  cost  price  and 
the  gain  per  cent?  How  do  you  find  the  loss  when  given  the 
cost  price  and  the  loss  per  cent  ? 

6.  I  sold  an  article  that  cost  $468  at  a  gain  of  8^%.  Find 
the  selling  price. 

7.  I  sold  an  article  that  cost  $  345  at  a  loss  of  331%.  Find 
the  selling  price. 

8.  When  given  the  cost  price  and  the  gain  or  loss  per  cent, 
how  do  you  find  the  selling  price  ? 

9.  The  gain  on  selling  an  article  that  cost  $450  was  7%. 
Find  the  selling  price. 

10.   Find  the  selling  price  of  an  article  which  cost  $  600  and 
was  sold  at  a  loss  of  5%. 


238  ARITHMETIC 

186.  A  speculator  bought  a  house  for  $  2349  and  sold  it  at 
a  gam  of  17%.     Find  the  selling  price. 

In  this  question  the  selling  price  is  the  sum  of  the  cost,  which  is  known, 
and  the  gain,  which  is  unknown.  The  gain  is  measured  by  the  number  17  % 
or  .17,  and  the  cost  price  $2349. 

$2349  cost 
.17  number 


16443 
2349 


$399.33  gain 

The  gain  =  17  %  of  $  2349  =  $  399.33. 

.-.  the  selling  price  =  $2349  +  $399.33  =  $2748.33. 

Exercise  151 

1.  Write  as  decimals  :  17%;  13%;  37%;  23^%;  146%;  346%; 

6%;8%;8i%;li%;i%;i%. 

2.  Find  34%  of  $893;  19%  of  643. 

3.  Find  27%  of  6594  bu.  of  wheat;  31%  of  1954. 

4.  If  39%  of  a  cargo  of  flour,  consisting  of  8492  bbl.,  was 
damaged,  how  many  barrels  were  damaged  ? 

5.  A  farmer  who  sold  his  crop  of  wheat  in  1899  for  $  967.20, 
received  13%  less  the  next  year.  How  much  less  did  he  receive 
for  his  crop  in  1899  than  in  1900  ? 

6.  A  grain  dealer  invested  $6459  in  wheat,  and  23%  of  that 
amount  in  oats.     How  much  did  he  invest  in  oats  ? 

7.  What  does  a  bill  for  $  1896  become  after  a  reduction  of  3%? 

8.  What  is  the  selling  price  of  an  article  costing  $18,  and 
sold  at  a  loss  of  9%? 

9.  What  is  the  selling  price  of  an  article  costing  $7,  and  sold 
at  a  gain  of  7  %  ? 


PERCENTAGE  239 

187.  Express  |  /o  as  a  decimal  and  also  as  9,  common  frac- 
tion in  its  lowest  terms. 

r/o=-oof. 

^^•^      100      500      125 

Exercise  152 

Express  as  decimals  and  also  as  common  fractions  in  their 
lowest  terms : 

1-  Wo ;  Wo ;  Wo ;  i% ;  f % ;  1% ;  1% ;  Wo- 

2.  Wo;  Wo',  Wo',  Wo',  Wo',  Wo',  Wo',  Mo- 

3.  1^% ;  1^% ;  tWo  ',  iWo ',  -iWo ',  \i% ;  A% ;  «%• 

4.  What  part  of  1%  is  i%  ?     f  %  ?     f  %  ?    ^%  ? 
What  is  1%  of  $800?    i%?    |%?    |%? 

5.  If  ^%  is  charged  for  sending  money  from  Chicago  to  Kew 

York,  what  is  charged  for  sending  $  1200  ? 

6.  If  1%  is  charged  for  sending  money  to  St.  Louis,  find  how 

much  is  charged  when  $  1632  is  sent. 

7.  If  i%  is  charged  as  commission  for  buying  stock,  what  is 

the  commission  on  buying  $  2400  stock  ? 

8.  If  1%  is  charged  for  selling  stock,  find  the  commission 

charged  for  selling  f  1600  stock. 

9.  6%  per  annum  is  what  per  cent  for  1  month  ?    f  %  a  month 

is  what  per  cent  per  annum  ? 

10.    What  is  the  cost  of  insuring  550  bbl.  of  flour,  worth  $  4  per 
barrel,  the  cost  of  insurance  being  i%  of  the  value  of  the  flour? 

188.  (1)  Express  102lfo  as  a  decimal. 

102|  0/^  =  IM  =  i.02|  =  1.025. 


240  ARITHMETIC 

(2)  I  send  my  agent  §5100,  which  is  102^  of  the  money 
which  he  invested  for  me  in  cotton.  What  does  my  agent 
pay  for  cotton  ? 

102  %  or  1.02  of  the  cost  of  the  cotton  =  $  5100. 
.'.  the  cost  of  the  cotton  =  $  5100  -^  1.02  =  $  5000. 

Exercise  153 

1.  Express  as  decimals :  103%;  104%;  J01%;  102%. 

2.  Express  as  decimals;  97%, ;  96%  ;  99%, ;  98%. 

3.  Express  as  decimals  :  1031%,  ;  104^%  ;  102f  %  ;  101^%,, 

4.  Express  as  decimals :  97^%;  96|%  ;  99J%  ;  98f%. 

5.  A  real  estate  broker  in  St.  Louis  received  $5047,  which 
was  103  %  of  the  sum  he  was  to  invest  in  real  estate.  What  sum 
did  he  invest  in  real  estate  ? 

6.  My  agent  sent  me  $3395,  which  was  97%  of  the  selling 
price  of  some  railway  stock.     What  did  the  stock  sell  for  ? 

7.  My  agent  on  selling  a  quantity  of  wheat  sent  me  98|%  of 
the  proceeds.     If  I  received  $  3546,  what  did  the  wheat  sell  for  ? 

8.  I  sent  my  agent  $4545.30,  which  was  104J%  of  the  sum 
he  was  to  invest  in  buying  silk.     What  did  he  pay  for  the  silk  ? 

189.  (1)  A  house  is  sold  for  116,400,  and  25^  of  the 
purchase  money  is  paid  down,  the  balance  to  remain  on 
mortgage.     How  much  remains  on  mortgage  ? 

In  this  problem  we  are  given  the  measured  whole,  i.e.  the  selling  price, 
and  the  number  or  25  %  of  it.  We  are  required  to  find  the  balance  which  is 
the  difference  between  the  selling  price  and  the  sum  paid. 

The  sum  paid  =  26  %  or  ^  of  $  16,400  =  $4100. 

.*.  the  balance  =  $  16,400  -  $  4100  =  $  12,300. 


PERCENTAGE  241 

(2)  On  Jan.  10  a  merchant  buys  goods  invoiced  at  1876.40, 
on  the  following  terms  :  4  months,  or  less  6%  if  paid  within 
10  days.     What  sum  will  pay  the  debt  on  Jan.  15  ? 

Since  Jan.  15  is  less  than  10  days  after  Jan.  10,  the  sum  due  will  be  6% 
less  than  $876.40. 

The  discount  =  6%  or  .06  of  $876.40  =  $ 52.584. 
.'.  the  sum  due  =  $  876.40  -  $  52.58  =  $  823.82. 

Exercise  154 

1.  A  maltster  malts  7200  bu.  of  barley,  which  in  the  process 
increases  12^  % .     How  many  bushels  of  malt  has  he  ? 

2.  Certain  books  are  bought  at  $1.75  each.  At  what  must 
they  be  sold  to  gain  12  %  ? 

3.  A  merchant  asked  30  %  advance  on  goods  which  cost  $  120, 
but  finally  took  25  %  less  than  the  price  asked.  What  did  he  sell 
them  for  ? 

4.  A  merchant  bought  apples  at  60  f^  per  bushel,  and  sold  them 
at  a  gain  of  25  %.  Find  the  selling  price  per  bushel.  How  many 
bushels  did  he  sell  if  he  received  all  together  $37.50? 

5.  Bought  $64  worth  of  apples  at  80^  per  bushel,  part  of 
which  being  damaged  and  rendered  worthless,  I  sold  the  remainder 
at  an  advance  of  50%,  receiving  $76.80.  How  many  bushels  were 
damaged  ? 

6.  If  10%  of  an  army  of  23,400  men  were  slain  in  battle,  and 
5%  of  the  remainder  were  mortally  wounded,  find  the  sum  of  the 
killed  and  mortally  wounded. 

7.  The  population  of  a  town  of  64,000  inhabitants  increases 
at  the  rate  of  2J%  in  each  year.  Find  its  population  1,  2,  and 
3  years  hence. 

8.  The  population  of  a  city  is  a  million;  it  increases  1^%  for 
3  years  successively.     Find  the  population  at  the  end  of  3  years. 

9.  A  lawyer  collected  $287.50,  and  charged  5%  for  his  ser- 
vices.    How  much  did  he  retain  and  how  much  did  he  pay  over  ? 


242  ARITHMETIC 

10.  .The  cost  price  of  a  book  is  ^1.60,  the  expense  of  sale  5% 
upon  the  cost  price,  and  the  profit  25%  upon  the  whole  outlay. 
Pind  the  selling  price  of  the  book. 

11.  The  cattle  on  a  certain  stock  farm  increase  at  the  rate  of 
18|%  per  annum.  If  there  are  4096  cattle  in  1899,  how  many 
will  there  be  in*1901  ? 

12.  A  man  bought  a  store  and  contents  for  $4720;  he  sold  the 
same  for  12|%  less  than  he  gave,  and  then  lost  15%  of  the  sell- 
ing price  in  bad  debts.     Find  his  entire  loss. 

13.  A  person  having  bought  goods  for  $40  sells  half  of  them  at 
a  gain  of  5%.  For  how  much  must  he  sell  the  remainder  so  as 
to  gain  20%  on  the  whole  ? 

14.  A  grocer  mixes  two  kinds  of  tea  which  cost  him  38^  and 
42  ^  per  pound  respectively  in  equal  quantities.  What  must  be 
the  selling  price  of  the  mixture  in  order  that  he  may  gain  15%  on 
his  outlay  ? 

15.  A  grain  dealer  expended  $2150  in  the  purchase  of  wheat, 
one-half  as  much  again  in  the  purchase  of  barley,  and  twice  as 
much  in  the  purchase  of  corn ;  he  sold  the  wheat  at  a  profit  of 
6%,  the  barley  at  a  loss  of  5%,  and  the  corn  at  a  gain  of  2%. 
Find  his  gain  on  the  whole  transaction, 

16.  A  persott  gave  $150  for  one  horse  and  $225  for  another. 
He  gold  the  first  horse  at  a  gain  of  20%,  and  the  second  at  a  loss 
of  20%.  Find  the  selling  price  of  each  horse  and  the  gain  or  loss 
on  the  whole  transaction. 

17.  A  sells  goods  to  B  which  cost  him  $465,  at  a  gain  of  6%, 
B  sells  them  to  C  at  a  loss  of  3%,  and  C  sells  them  to  D,  gaining 
10%.     What  did  D  give  for  the  goods  ? 

18.  A  man  having  bought  a  lot  of  goods  for  $450  sells  \  at 
a  loss  of  5%,  ^  at  a  gain  of  7%,  and  the  remainder  at  a  gain  of 
2%.     Find  the  total  gain. 

19.  A  merchant  began  business  with  a  capital  of  $30,000.  He 
gained  163%  the  first  year,  which  he  added  to  his  capital,  and 


PERCENTAGE  243 

12i^%  the  second  year,  which  he  added  to  his  capital.  In  the 
third  year  he  lost  20%.  Find  his  capital  at  the  end  of  the  third 
year. 

20.  Sugar  being  composed  of  49.856%  of  oxygen,  43.265%  of 
carbon,  and  the  remainder  hydrogen,  find  how  many  pounds  of 
each  of  these  materials  there  are  in  1  T.  of  sugar. 

21.  A  merchant  buys  a  bill  of  dry  goods,  April  16,  amounting 
to  $6377.84,  on  the  following  terms:  4  mo.,  or  less  5%  if  paid 
within  30  da.     How  much  would  settle  the  account  on  May  15  ? 

22.  Water  is  composed  of  88.9%  of  oxygen  and  11.1%  of 
hydrogen.  HoW  many  pounds  are  there  of  each  in  1  cu.  ft.  of 
water  ?     (A  cubic  foot  of  water  weighs  1000  oz.) 

190.  I  sold  an  article  which  cost  1 64  at  a  gain  of  $  24. 
Find  my  gain  per  cent. 

The  gain  per  cent  =  ||  or  |  or  37 1%  of  the  cost. 

Exercise  155 

1.  I  sold  an  article  which  cost  $  75  at  a  gain  of  $25.  Find  the 
gain  per  cent. 

2.  A  farmer  had  200  A.  and  sold  50  A.  Find  what  per  cent  of 
his  farm  he  sold. 

3.  I  paid  $225  for  a  horse,  and  sold  it  at  a  profit  of  $45. 
Find  my  gain  per  cent. 

4.  I  lost  $  36  on  selling  an  article  which  cost  $  108.  Find  the 
loss  per  cent. 

5.  What  two  things  do  you  compare  to  find  the  gain  per  cent  ? 
The  loss  per  cent  ? 

6.  A  suburbanite  earns  $  125  a  month,  and  pays  $  5  a  month  for 
a  monthly  railway  ticket.  What  per  cent  of  his  salary  does  his 
ticket  cost  ? 

191.  (1)  A  merchant  sold  60  yd.  of  cloth  from  a  web  con- 
taining 150  yd.     What  per  cent  of  the  web  did  he  sell  ? 


244  ARITHMETIC 

We  are  here  given  the  measured  part  and  the  measured  whole,  and  we  are 
required  to  find  the  number  expressing  the  ratio  of  the  part  to  the  whole. 
The  quantity  sold  =  /^  or  j  or  40  %  of  the  web. 

(2)  An  article  which  cost  13.60  was  sold  for  14.32.    Find 
the  gain  per  cent. 

The  gain  =  $4.32  -  $3.60  =  72  cents. 
.*.  the  gain  =  ^Vs  or  I  or  20  %  of  the  cost 

Exercise  156 

1.  The  cost  price  of  an  article  is  f  64,  and  the  gain  on  selling 
$  16.     Find  the  gain  per  cent. 

2.  Tea  is  bought  at  84^  per  pound  and  sold  at  98^.     Find  the 
gain  per  cent. 

3.  Out  of  48  eggs  6  were  broken.     What  per  cent  of  the  whole 
number  was  broken  ? 

4.  The  cost  price  of  an  article  was  $56,  and  the  selling  price 
$  49.     Find  the  loss  per  cent. 

5.  What  per  cent  is  1  in.  of  1  ft.  ?     1  ft.  of  1  yd.  ?     1  yd.  of 
1  rd.  ?     1  rd.  of  1  mi.  ? 

6.  What  per  cent  is  1  min.  of  1  hr.  ?     1  hr.  of  1  da.  ?     1  da. 
of  1  wk.  ?     1  wk.  of  1  yr.  ? 

7.  What  per  cent  is  1  pt.  of  1  qt.  ?     1  qt.  of  1  gal.  ? 

8.  What  per  cent  of  1  pk.  is  1  qt.  ?     Of  1  bu.  is  1  pk.  ? 

9.  A  merchant  by  selling  1  lb.  of  butter  gains  the  cost  price 
of  1  oz.     What  is  his  gain  per  cent  ? 

10.  1  lb.  Troy  is  what  per  cent  of  1  lb.  Avoirdupois  ?     1  lb. 
Avoirdupois  is  what  per  cent  of  1  lb.  Troy  ? 

11.  The  volume  of  1  gal.  is  what  per  cent  of  1  cu.  ft.  ?    1  cu.  ft. 
is  what  per  cent  of  1  gal.  ? 

12.  An  area  containing  1  sq.  yd.  is  increased  by  4  sq.  ft.     Find 
the  per  cent  of  increase. 

13.  If  £  1  is  worth  $  4.866,  what  per  cent  of  £  1  is  $  1  ? 


PERCENTAGE  245 

Exercise  157 

1.  A  paymaster  receives  $150,000  from  the  treasury,  but 
fails  to  account  for  $  2250.  What  is  the  percentage  of  loss  to  the 
government  ? 

2.  A  city  of  16,000  inhabitants  increases  in  a  given  time  to 
20,000.     Find  the  increase  per  cent. 

3.  $  640  increased  by  a  certain  per  cent  of  itself  equals  $  720. 
Eequired  the  rate  per  cent. 

4.  A  house  worth  $  3500  rents  for  $  420.  For  what  per  cent 
of  its  value  does  it  rent  ? 

5.  If  a  tradesman  gains  $  1.32  on  an  article  which  he  sells  for 
$  5.28,  what  is  his  gain  per  cent  ? 

6.  An  article  which  cost  84^  is  sold  for  93^.  Find  the  gain 
per  cent. 

7.  A  city  gained  2467  in  population  in  5  years.  If  its  popu- 
lation was  14,802  live  years  ago,  what  was  the  gain  per  cent  ? 

8.  In  a  certain  year  the  number  of  graduates  of  a  school  was 
70.  Ten  years  later  it  was  210.  Find  the  rate  per  cent  of 
increase. 

9.  A  tea  merchant  mixes  40  lb.  of  tea  at  45^  per  pound  with 
50  lb.  at  27  ^  per  pound,  and  sells  the  mixture  at  42  ^  per  pound. 
What  per  cent  profit  does  he  make  ? 

10.  Paid  f  80  freight  on  goods  that  cost  fll20.  What  must 
they  be  sold  for  to  make  a  profit  of  20%  on  the  full  cost  ? 

11.  A  grocer  uses  for  a  1-pound  weight  one  which  weighs  only 
15f  oz.     What  does  he  gain  per  cent  by  his  dishonesty  ? 

12.  I  bought  500  sheep  at  $  4  a  head ;  their  food  cost  me  f  1.50 
a  head ;  I  then  sold  them  at  $  6  a  head.     Find  my  gain  per  cent. 

13.  A  man's  income  is  derived  from  the  proceeds  of  f  4550  at 
a  certain  rate  per  cent,  and  $  5420  at  1  %  more  than  the  former 
rate.     His  whole  income  -being  $  453,  find  the  rates. 

14.  Coffee  is  bought  in  50-pound  bags  for  $  16  and  sold  for  36  ^ 
a  pound.     Find  the  rate  of  profit  per  cent. 


246  ARITHMETIC 

192.  The  gain  $  36  is  22 1%  of  the  cost ;  find  the  cost. 

The  gain  =  22|%  or  |  of  the  cost. 
I  of  the  cost  =  $  36 
.-.the  cost  =  $  36  X  I  =  $  162. 

Exercise  158 

1.  The  gain  $  25  is  12}%  of  the  cost.     Find  the  cost. 

2.  The  gain  ^  36  is  37}%  of  the  cost.     Find  the  cost. 

3.  ^  18  is  6%  of  my  month's  salary.     What  is  my  salary  ? 

4.  8%  of  my  rent  is  equal  to  $  32.     What  rent  do  I  pay  ? 

5.  The  gain  f  60  is  12%  of  the  cost.     Find  the  cost. 

6.  The  selling  price  ^  3810  is  127%  of  the  cost.    Find  the  cost. 

7.  I  sold  an  article  for  $  92,  gaining  15%  of  the  cost.  Find  the 
cost. 

8.  The  selling  price  $5640  is  94%  of  the  cost.     Find  the  cost. 

9.  I  sold  an  article  for  $  720,  losing  4%  of  the  cost.  Find  the 
cost. 

193.  A  trader  sold  a  horse  at  an  advance  of  12%,  gaining 
$  18.     Find  the  cost  price  of  the  horse. 

12  %  or  ^%  of  the  cost  =  $  18. 
3^  of  the  cost  =  $6. 
f  ^  of  the  cost  =  $  160. 
.-.  the  C08t  =  !$150. 

Exercise  159 

1.  A  quantity  of  sugar  was  sold  at  an  advance  of  12J%.  If 
the  gain  was  $  17,  what  was  the  cost  ? 

2.  Cloth  was  sold  at  a  loss  of  37}%.  If  the  loss  was  36^  a 
yard,  what  was  the  cost  per  yard  ? 

3.  Forty-five  per  cent  of  a  piece  of  cloth  was  sold.  If  135  yd. 
were  sold,  how  many  yards  were  in  the  piece  at  first?  How 
many  remained  unsold  ? 


PERCENTAGE  247 

4.  3%  more  is  gained  by  selling  a  horse  for  ^333  than  by 
selling  him  for  $  324 ;  find  his  original  price. 

5.  A  man  bought  a  horse  which  he  sold  at  a  loss  of  8%.  If 
he  had  received  $  24  more,  he  would  have  gained  7%.  What  did 
the  horse  cost  him  ? 

6.  A  clerk  pays  16%  of  his  salary  each  year  for  board.  If 
his  board  costs  him  $  208  a  year,  what  is  his  salary  ? 

7.  A  man  sold  a  field  consisting  of  15  A.,  which  was  6J%  of 
his  farm.     How  many  acres  were  in  the  farm  at  first  ? 

8.  25%  of  my  money  is  invested  in  bank  stock  and  the 
remainder  in  business ;  what  per  cent  of  my  money  is  invested 
in  business  ?  My  bank  stock  is  worth  what  part  of  my  business 
capital  ?  If  I  have  $  4800  invested  in  business,  what  is  the  value 
of  my  bank  stock  ? 

9.  Twenty  per  cent  of  my  money  is  invested  in  business,  and 
the  remainder,  which  is  $  12,800,  in  real  estate.  How  much  have 
I  invested  in  business  ? 

10.  I  invested  25%  of  my  money  in  business,  and  put  6|%  of 
the  remainder  in  the  bank.  If  I  put  $  600  in  the  bank,  how 
much  money  did  I  have  at  first  ? 

11.  Twenty -eight  per  cent  of  a  sum  of  money  was  invested  in 
business,  and  12^%  of  the  remainder  in  real  estate.  If  the  sum 
invested  in  business  exceeds  that  invested  in  real  estate  by  $  1900, 
find  the  amount  of  money  I  had  at  first. 

194.  (1)  A  man  invests  77|  %  of  his  capital  in  bank  stock, 
and  has  129,367  left.     What  "is  his  capital  ? 

The  amount  left  =  100  %  -  77^  %  or  22|  %  of  his  capital. 
22^  %  or  5%  of  his  capital  =  $  29,367. 
^\  of  his  capital  =  $  3263. 
|§  of  his  capital  =  $  130,520. 
.-.  his  capitals  $130,520. 


248  ARITHMETIC 

(2)  A  profit  of  17  ^  is  made  by  selling  an  article  at  an 
advance  of  $  24.50.  What  would  have  been  the  selling  price 
if  the  loss  had  been  8  ^  ? 

17  %  of  the  cost  =  $  24.50. 
1  %  of  the  cost  =  $1.4412. 
92  %  of  the  cost  =  $  132.59. 
.*.  the  second  selling  price  =  $  132.59. 

Exercise  160 

1.  A  man  invests  42%  of  his  capital  in  real  estate,  and  has 
$  53,070  left.     What  is  his  capital  ? 

2.  A  bankrupt's  assets  are  $  23,625,  and  he  pays  62J%  of  his 
debts.     How  much  does  he  owe  ? 

3.  A  merchant  loses  6J%  of  the  cost  price  by  selling  an 
article  at  a  loss  of  ^  27.30.  Find  the  cost  price,  and  also  at  what 
he  must  sell  it  to  gain  7^%. 

4.  By  selling  a  house  at  a  loss  of  $  150,  a  real  estate  dealer 
loses  6|%  of  the  cost.  Find  the  cost  and  also  the  gain  per  cent 
if  it  had  been  sold  for  $  2625. 

5.  I  sold  a  lot  at  a  gain  of  8J%,  thereby  gaining  $  113.  What 
should  I  have  sold  it  for  to  gain  9%  ? 

6.  Coals  are  20%  cheaper  this  year  than  last.  If  the  price 
were  to  rise  $  1  per  ton,  they  would  still  be  50  ^  per  ton  cheaper 
than  last  year.     Find  the  price  last  year. 

7.  A  person  asked  for  a  lot  of  land  40%'  more  than  it  cost 
him,  but  finally  reduced  his  price  15%,  gaining  on  the  whole 
$  380.     How  much  did  the  land  cost  him  ? 

8.  A  merchant  sold  J  of  a  quantity  of  cloth  at  a  gain  of  20%, 
and  the  remainder  at  cost.  His  gain  was  what  per  cent  of  the 
cost  ?    If  he  gained  $  7.29,  what  was  the  cost  of  the  goods  ? 

9.  A  merchant  sold  |  of  a  quantity  of  tea  at  a  gain  of  12%, 
and  the  remainder  at  a  gain  of  9%,  gaining  all  together  $2.75. 
Find  the  cost  of  the  tea. 


PERCENTAGE  249 

10.  A  speculator  gained  20%  on  |  of  his  investment,  and  lost 
24%  on  the  remainder.  All  together  he  made  $  270.  Find  the 
amount  of  his  investment. 

11.  Ten  per  cent  of  an  army  were  slain  on  the  field  of  battle, 
and  5  per  cent  of  the  remainder  were  mortally  wounded.  The 
difference  between  the  killed  and  mortally  wounded  w^as  1100. 
How  many  men  went  into  battle  ? 

195.  (1)  A  horse  was  sold  for  $117,  which  was  S^fo  more 
than  it  cost.     Find  the  cost  price. 

The  gain  =  8^  %  or  ^^  of  the  cost. 
The  selling  price  =  ^|  of  the  cost. 
^1  of  the  cost  =  $  117. 

.-.  the  cost  =  $  117  X  ^1  =  $108. 

(2)  A  horse  was  sold  for  1154,  which  was  81/0  less  than 
it  cost.     Find  the  cost  price. 

The  loss  =  8|  %,  or  ^^  of  the  cost. 
The  selling  price  =  ^^  of  the  cost. 
^1  of  the  cost  =  $  154. 

.-.  the  cost  =  $  154  X  H  =  $  168. 

Exercise  161 

1.  A  house  and  lot  were  sold  for  $  3600,  which  was  20%  more 
than  they  cost.     Find  the  cost  price. 

2.  A  house  and  lot  were  sold  for  $  4200,  which  was  25%  less 
than  they  cost.     Find  the  cost  price. 

3.  A  speculator  gained  7%  by  selling  wheat  for  $  2140.  Find 
the  cost  price. 

4.  Eggs  are  sold  at  the  rate  of  15  ^  per  dozen,  a  profit  of  25% 
being  made.     What  is  the  cost  price  per  dozen  ? 

5.  I  sold  a  book  for  42^,  gaining  16|%.  Find  the  cost  price. 
How  much  would  be  gained  by  selling  at  a  gain  of  25%  ?  What 
would  then  be  the  selling  price  ? 


250  ARITHMETIC 

6.  By  selling  hats  at  60^  each,  a  merchant  gains  33|%.  Find 
the  cost  price.  What  would  have  been  the  actual  loss,  and  what 
the  loss  per  cent,  if  they  had  been  sold  at  36^  each  ? 

7.  I  sold  a  lot  of  land  for  $  600,  thereby  gaining  20%.  Find 
the  cost  price. 

8.  I  sold  a  lot  of  land  for  $  600,  thereby  losing  20%.  Find 
the  cost  price. 

9.  What  is  the  cost  of  both  lots  in  questions  7  and  8  ?  What 
is  their  selling  price  ?  How  much  is  the  cost  of  both  greater  than 
their  selling  price  ? 

10.  A  dealer  sold  two  bicycles  for  $45  each,  losing  25%  on 
one  and  gaining  25%  on  the  other.  How  much  did  he  lose  on 
the  whole  transaction  ? 

196.  (1)  If  a  debt,  after  a  reduction  of  8J&,  becomes 
$1008.80,  what  would  it  become  after  a  reduction  of  456  ? 

After  a  reduction  of  3  %,  the  amount  owed  =  97  %  of  the  original  debt, 
and  after  a  reduction  of  4  %  it  becomes  90  %  of  the  original  debt. 

97  %  of  the  debt  =  $  1008.80. 

1  %  of  the  d,ebt  =  $  10.40. 

96  %  of  the  debt  =  $  998.40. 

/.  after  a  reduction  of  4  %  the  debt  =  $  998.40. 

(2)  The  population  of  a  city  increases  2^  yearly.  It  now 
has  132,651  inhabitants.  How  many  had  it  1,  2,  and  8  years 
ago? 

The  population  now  =  102%,  or  1.02  of  that  1  year  ago. 
1.02  of  the  population  1  year  ago  =  132,651. 

.-.  the  population  1  year  ago  =  132,651  -4- 1.02  =  130,050. 
.-.  the  population  2  yeare  ago  =  130,050  -r-  1.02  =  127,500. 
.-.  the  population  3  years  ago  =  127,600  -r- 1.02  =  125,000. 
Prove  these  answers  correct. 


PERCENTAGE  251 

Exercise  162 

1.  A  liorse  was  sold  for  $658,  which  was  16|%  more  than 
its  cost.     How  much  did  it  cost  ? 

2.  A  speculator  gained  3%  by  selling  wheat  for  $6437.50. 
Pind  the  cost  price. 

3.  A  mercha.nt,  after  a  business  of  five  years,  found  his  capi- 
tal increased  to  $28,000,  showing  a  gain  of  60%  on  his  original 
capital.     Find  that  capital. 

4.  Eggs  are  sold  at  the  rate  of  5  for  6^,  a  profit  of  20% 
being  made.     Eind  the  price  at  which  they  are  bought. 

5.  In  1901  a  city  has  a  population  of  28,000  inhabitants.  If 
its  population  increased  17||%  in  the  two  years  previous,  what 
was  it  in  1899  ?  If  its  population  decreased  17|^%  in  the  two 
years  previous,  what  was  it  in  1899  ? 

6.  By  selling  an  article  for  $2.64  a  merchant  loses  12%. 
What  was  the  cost,  and  for  what  must  he  sell  it  to  gain  16|%  ? 

7.  A  merchant  sells  tea  at  75^  per  pound,  thereby  losing  5%. 
What  was  the  cost,  and  at  what  price  per  pound  must  it  be  sold 
to  gain  41%  ? 

8.  Flour  is  sold  for  $4.80  per  barrel,  at  a  loss  of  20%.  What 
selling  price  would  give  20%  gain  ? 

9.  By  selling  an  article  for  $23,  8%  is  lost.  What  per  cent 
is  gained  if  it  is  sold  for  $  31  ? 

10.  I  sold  goods  at  $21.60  per  hundredweight,  thereby  gaining 
14|^%.     Find  the  cost  per  pound. 

11.  A  farmer  sold  his  crop  of  wheat  in  1871  for  8%  more  than 
he  obtained  for  his  crop  of  the  preceding  year ;  he  received  for 
both  crops  $  2600 ;  how  much  did  he  get  for  his  crop  of  1870  ? 

12.  I  sold  two  houses,  receiving  $2400  for  each.  On  the  first 
I  gained  25%,  and  on  the  second  lost  25%.  Find  my  loss  on 
both  transactions. 

13.  I  sold  a  lot  of  land  for  $600,  thereby  gaining  20%  ;  a 
second  for  $600,  losing  20%.     Find  my  loss  on  both  transactions. 


252  ARITHMETIC 


PROFIT   AND  LOSS 


197.  The  Profit  is  the  amount  by  which  the  selling  price 
exceeds  the  buying  price. 

The  rate  of  profit  is  usually  expressed  as  a  certain  per  cent 
of  the  cost  price. 

The  Loss  is  the  amount  by  which  the  selling  price  falls 
short  of  the  cost  price. 

The  rate  of  loss  is  usually  expressed  as  a  certain  per  cent 
of  the  cost  price. 

198.  (1)  At  a  forced  sale  a  bankrupt's  house  was  sold  for 
iSOOO,  which  was  20%  less  than  its  real  value.  If  the  house 
had  been  sold  for  $  12,000,  what  per  cent  of  its  real  value 
would  it  have  brought  ? 

80%  of  the  value  of  the  house  =  $  8,000. 

1  %  of  the  value  of  the  house  =         100. 

100%of  the  value  of  the  house  =    10,000. 

.*.  the  second  selling  price  =  HM^  of  the  value 
=  120%  of  the  value. 

(2)  The  manufacturer  of  an  article  makes  a  profit  of  25%, 
the  wholesale  dealer  makes  a  profit  of  20%,  and  the  retail 
dealer  makes  a  profit  of  30%.  What  is  the  cost  to  the 
manufacturer  of  an  article  that  retails  at  $15.60? 

Let  the  cost  to  the  manufacturer  =100  units  of  money. 

The  selling  price  of  the  manufacturer      =  125  units  of  money. 
The  gain  of  the  wholesale  dealer  =    25  units  of  money. 

The  selling  price  of  the  v^^holesale  dealer  =  150  units  of  money. 
The  gain  of  the  retail  dealer  =    45  units  of  money. 

The  selling  price  of  the  retail  dealer         =  195  units  of  money. 

195  units  of  money  =  $  16.60. 
1  unit  of  money  =        .08. 

100  units  of  money  =      8.00. 
.*.  the  prime  cost  =      8.00. 


PROFIT  AND   LOSS  253 

GENERAL   STATEMENT   OF   SOLUTION 

(3)  Represent  the  cost  to  the  manufacturer  by  100  units 
of  money,  and  then  find  the  number  of  units  representing 
respectively  the  selling  prices  of  the  manufacturer,  the  whole- 
sale dealer,  and  the  retail  dealer.  Put  the  number  of  units 
of  money  which  represent  the  retail  price  equal  to  $15.60 
and  find  the  value  of  100  units  of  money,  which  is  the  cost 
of  manufacturing. 

QUESTION  IN  PROOF 

(4)  The  manufacturer  of  an  article  makes  a  profit  of  25^, 
the  wholesale  dealer  a  profit  of  20^,  and  the  retail  dealer  a 
profit  of  30^.  What  is  the  retail  price  of  an  article  which 
cost  the  manufacturer  18? 

Proof 
The  manufacturer's  gain  =  25  %  of  $  8  =  $  2. 

The  manufacturer's  selling  price      =  $  10. 
The  wholesale  dealer's  gain  =  20  %  of  $  10  =  $  2. 

The  wholesale  dealer's  selling  price  =  $  12. 
The  retail  dealer's  gain  =  30  %  of  $  12  =  $3.00. 

The  retail  dealer's  selling  price         =  $  15.60. 
.-.  $8  is  the  correct  answer  to  the  previous  question, 

MAKING   QUESTIONS 

(5)  Make  a  question  in  which  you  are  given  the  selling 
price  and  the  gain  per  cent,  and  are  required  to  find  the  cost 
price. 

Making 

Let  the  cost  of  a  house  =  $  6250. 

Let  the  gain  per  cent  on  selling  =  37^  %. 

The  gain  =  37i  %  of  $  6250  =  $  2343.75. 

The  selling  price  =  $6250  +  $2343.75  =  $  8593.75. 


254  ARITHMETIC 


Problems 


A  house  was  sold  at  37|^  above  cost.  If  the  selling  price 
was  18593.75,  find  the  cost  price. 

Other  questions  may  also  be  written  down  from  the  same 
making,  thus : 

A  house  which  cost  16250  was  sold  at  a  gain  of  37|^. 
Find  the  selling,  price. 

A  house  which  cost  $6250  was  sold  for  18593.75.  Find 
the  gain  per  cent. 

A  house  which  cost  16250  was  sold  at  a  gain  of  $2343.75. 
Find  the  gain  per  cent. 

In  the  following  exercise  state  in  general  terms  how  to 
solve  each  question.  Prove  some  of  your  answers  correct, 
framing  the  question  in  proof.  Make  questions  similar  to 
problems  in  the  exercise. 

Exercise  163 

1.  A  lot  of  dry  goods  was  sold  at  an  advance  of  18%.  If  the 
gain  was  1 436.50,  what  was  the  cost  ? 

2.  I  made  a  mixture  of  wine  consisting  of  1  gal.  at  50^,  3  at 
90^,  4  at  ^1.20,  and  12  at  40^.  I  sell  the  mixture  at  80^  a  gal- 
lon.    Find  my  gain  per  cent. 

3.  A  merchant's  price  is  25%  above  cost.  If  he  allows  a  cus- 
tomer a  discount  of  12%  on  his  bill,  what  per  cent  profit  does  he 
make? 

4.  Cloth  when  sold  at  a  loss  of  16|%  brings  $2.50  a  yard. 
What  would  be  the  gain  or  loss  per  cent  if  sold  at  $4  a  yard  ? 

5.  Eggs  are  bought  at  25  i  a  dozen,  and  sold  at  the  rate  of  8  for 
20^.     Find  the  rate  of  profit. 

6.  A  merchant  sells  goods  to  a  customer  at  a  profit  of  60%, 
but  the  buyer  becomes  bankrupt  and  pays  only  70  cents  on  the 
dollar,    What  per  cent  does  the  merchant  gain  or  lose  on  the  sale  ? 


PROFIT   AND   LOSS  255 

7.  A  man  bought  a  horse  which  he  sold  again  at  a  loss  of 
10%.  If  he  had  received  $45  more  for  him  he  would  have  gained 
121%.     Find  the  cost  of  the  horse. 

8.  A  bookseller  sold  a  book  at  17%  below  cost,  but  had  he 
charged  48  cents  more  for  it,  he  would  have  gained  7%.  Find 
the  cost  of  the  book  to  the  bookseller,  and  the  price  at  which  he 
sold  it. 

9.  A  tradesman  bought  goods  for  $  1200  and  sold  ^  of  them  at 
a  loss  of  10%.  For  how  much  must  he  sell  the  remainder  to  gain 
20%  on  the  whole  ? 

10.  A  man  bought  a  house  and  lot  for  f  4750.  After  spending 
$11-22  on  repairs  and  improvements,  and  paying  $128  for  taxes 
and  other  expenses,  he  sold  the  property  for  $  6400.  What  rate 
per  cent  of  profit  did  his  investment  yield  him  ? 

11.  By  selling  cloth  at  $1.20  per  yard,  a  tradesman  lost  6J% 
on  his  outlay.     At  what  price  must  he  sell  it  to  gain  12i~%  ? 

12.  If  a  manufacturer  sells  an  article  of  which  the  first  cost  is 
$400,  to  a  wholesale  dealer  at  10%  profit,  the  wholesale  dealer 
to  the  retailer  at  15%  profit,  and  the  retailer  to  the  consumer  at 
30%  profit,  what  sum  is  paid  by  the  consumer  as  profits  in  addi- 
tion to  the  first  cost  of  the  article  ? 

13.  A  grocer  sold,  at  51^  per  pound,  a  portion  of  a  stock  of  tea, 
incurring  a  loss  of  15%  and  a  total  loss  of  $18  on  the  quantity 
sold.     How  many  pounds  did  he  sell  ? 

14.  A  merchant  marks  his  goods  so  that  he  may  allow  a  dis- 
count of  5%,  and  still  make  a  profit  of  15%.  Find  the  marked 
price  of  broadcloth  that  cost  him  $  3.80  a  yard. 

15.  A  person  sold  two  horses  at  $160  each,  losing  20%  on 
one  and  gaining  20%  on  the  other.  Did  he  gain  or  lose  on  the 
whole  transaction,  and  how  much  ? 

16.  A  speculator  paid  $1400  for  two  lots,  the  price  of  one  of 
them  being  40%  that  of  the  other.  He  sold  the  cheaper  lot  at  a 
gain  of  50%,  and  the  dearer  one  at  a  loss  of  30%.  Find  his  gain 
or  loss  per  cent  on  the  whole  transaction. 


256  ARITHMETIC 

COMMERCIAL  OR  TRADE  DISCOUNT 

199.  Commercial  discount  is  an  allowance  made  by  mer- 
chants upon  their  catalogue  prices. 

The  commercial  discount  is  reckoned  at  a  certain  rate  per 
cent. 

Sometimes  several  discounts  are  allowed  to  a  purchaser. 

In  such  a  case,  the  first  discount  is  to  be  deducted,  and 
then  the  second  discount  is  to  be  reckoned  upon  the  re- 
mainder and  then  deducted,  and  so  on  for  each  successive 
discount. 

200.  What  is  the  net  amount  of  a  bill  for  $  720  subject  to 
discounts  of  20%  and  6%  ?  Find  a  single  discount  equiva- 
lent to  these  successive  discounts. 

The  first  discount  =20%  of  $720  =  $144. 
The  first  remamder  =  $  720  -  $  144  =  $576. 
The  second  discount  =  6%  of  $576  =  $34.56. 
.-.  the  net  amount  =  $576  -  $34.56  =  $541.44. 
Again,  the  single  equivalent  discount  =  $720  -  $541.44  =  $178.56. 

.-.  the  rate  of  a  single  discount  =  $178.56  ^  $720  =  .248  =  24.8%. 
Why  is  the  single  discount  24.8%  less  than  the  sum  of  the  two  discounts  ? 

Exercise  164 

1.  An  invoice  was  f  650,  trade  discounts  20%  and  8%  oft*. 
Find  the  cost  of  the  goods. 

2.  What  is  the  net  amount  of  a  bill  of  goods,  the  list  price  of 
which  is  $245,  trade  discounts  18%  and  5%  off  for  cash? 

3.  What  is  the  difference  on  an  invoice  of  $540,  between 
40%  direct  discount,  and  discounts  of  25%  and  15%  ? 

4.  A  dealer  buys  a  book,  list  price  80^,  at  a  discount  of  25%  ; 
he  sells  the  book  for  80^.     What  per  cent  is  the  profit? 


COMMERCIAL   OR  TRADE   DISCOUNT 


257 


5.  What  is  the  net  amount  of  a  bill  of  f  480,  discounts  being 
121%  and  8%  ?  Find  a  single  discount  equivalent  to  these  suc- 
cessive discounts. 

6.  A  man  bought  goods  at  discounts  of  20%  and  5%.  The 
list  price  was  $400.     Eind  what  he  paid  for  the  goods. 

7.  A  dealer  bought  goods  at  15%  and  12%  off.  The  list 
price  of  the  goods  was  $  250.     Find  what  he  paid  for  the  goods. 

8.  Find  the  net  cash  amount  of  a  bill  for  $1266,  subject  to 
discounts  of  331%,  10%,  and  5%,  for  cash. 

9.  Find  the  difference  between  a  single  discount  of  40%,  and 
successive  discounts  of  30%  and  10%. 

10.  Find  the  net  amount  of  a  bill  of  $250,  discounts  being  30, 
16,  and  6. 

11.  Find  the  net  cash  amount  of  a  bill  of  $  256,  discounts  being 
25%,  12^%,  5%.  Find  a  single  discount  equivalent  to  these 
three  successive  discounts. 

12.  In  the  following  examples,  in  which  the  list  prices  and  the 
rates  of  discount  are  given,  find  the  cost  prices : 


List  Price 

Rates  of  Discount 

a. 

$125.00    .     .     . 

.    .    .    15%,     10% 

b. 

112.50     .     .     . 

.    .    .    30%,    25% 

c. 

.3147.00     .     .     . 

.    .     .    45%,    20%,    5% 

d. 

796.00     .     .     . 

.     .    .    35%,    121%,  8% 

e. 

2378.00     .     .     . 

.    .    .     331%,  15%,    6% 

/. 

432.75     .     .     . 

.    .    .    50%,    10%,    6% 

13.  A  merchant  who  receives  successive  discounts  of  20%, 
15%,  and  10%,  on  a  bill  of  $750,  sells  at  an  advance  of  33i%. 
What  does  he  sell  his  goods  for? 

14.  What  is  the  difference  between  discounting  a  bill  of  $3000 
at  40%,  and  then  taking  a  discount  off  the  remainder  of  5%  for 
cash,  and  discounting  the  whole  at  45%  ? 


258  ARITHMETIC 

15.  A  merchant  buys  goods  at  40  and  20  off  the  list  price  and 
sells  them  at  30  and  10  off  the  list  i^rice.  What  is  his  gain  per 
cent? 

16.  An  invoice  of  crockery,  amounting  to  f  1500,  was  sold 
Jan.  3,  at  90  days,  subject  to  40%  and  10%  discount,  with  an 
additional  discount  of  G%  if  paid  within  20  days.  How  much 
will  be  required  to  pay  the  bill  on  Jan.  21  ? 

COMMISSION  AND  BROKERAGE 

201.  A  Commission  Merchant  is  one  who  buys  or  sells 
goods  for  other  persons  by  their  authority.  Commission 
merchants  are  usually  placed  in  possession  of  the  goods 
bought. 

A  Broker  is  a  person  who,  in  the  name  of  his  principal, 
effects  contracts  to  buy  or  sell. 

The  broker  is  not  in  general  placed  in  possession  of  the 
goods  bought  or  sold. 

The  title  Broker  is  also  applied  to  persons  who  deal  in 
stocks,  bonds,  bills  of  exchange,  promissory  notes,  etc.,  and 
to  mercantile  agents,  who  transact  the  business  for  a  ship  in 
port. 

Commission  is  the  charge  made  by  an  agent  for  transacting 
business. 

In  buying,  the  commission  is  reckoned  on  the  cost  price ; 
in  selling,  the  commission  is  reckoned  on  the  selling  price. 

202.  (1)  A  commission  merchant  sold  270  bbl.  of  flour  at 
16  a  barrel,  and  received  5%  commission.  What  was  his 
commission  ?     How  much  did  he  remit  to  his  employer  ? 

The  selling  price  =  270  x  $6  =  $1620. 
.-.  the  commission  =  6%  of  $1620  =  $81. 
.-.  the  amount  remitted  =  $1620  -  $81  =  $1539. 


COMMISSION   AND  BROKERAGE  259 

(2)  A  commission  of  $242.58  Avas  charged  for  selling 
#3772  worth  of  goods.     What  was  tlie  rate  of  commission  ? 

24'2  ')8 

The  commission  = '- —  of  the  selling  price 

3772  ^ 

=  .0043  of  the  selling  i)rice. 
.'.  the  rate  of  commission  =  6.43%. 

(3)  A  grain  dealer  charged  3|  %  for  selling  a  quantity  of 
wheat,  and  received  for  his  commission  $218.40.  For  how 
much  did  he  sell  the  wheat  ? 

The  commission  =  3|%  or  .035  of  the  selling  price. 
.035  of  the  selling  price  =  $218.40. 
.-.  the  selling  price  =  218.40  ^  .035  =  $6240. 

(4)  If  1512.50  include  the  price  paid  for  certain  goods, 
and  2|%  commission  to  the  agent,  how  much  money  does 
the  agent  expend  in  purchasing  the  goods  ? 

Let  the  cost  price  of  the  goods  =  100  units  of  money. 
Then  the  commission  =  2^  units  of  money. 
The  amount  sent  to  the  commission  merchant  =  102|  units  of  money. 
102^  units  =  $512.50. 

1  unit  =  512.50  ^  102.5  =  $5. 
100  units  =  $  500. 
.'.  the  cost  of  the  goods  =  $  500. 

As  in  Exercise  163  give  the  general  statements  of  solu- 
tions, prove  answers,  and  make  questions.  Do  this  also  in 
each  of  the  following  exercises  : 

Exercise  165 

1.  A  commission  merchant  sold  480  bbl.  of  flom-  at  $3.50  a 
V)arrel  on  a  commission  of  2%.  What  was  his  commission? 
How  much  did  he  remit  to  his  employer  ? 


260  ARITHMETIC 

2.  My  agent  sold  coffee  to  the  amount  of  $850  on  a  commis- 
sion of  3%.  Find  his  commission  and  also  the  amount  remitted 
to  his  employer. 

3.  An  agent  sold  210  bu.  of  oats  at  60^  a  bushel,  and  charged 
$  3.78  for  doing  so.     Find  his  rate  of  commission. 

4.  On  a  debt  of  $  1725  a  creditor  receives  a  dividend  of  60%, 
on  which  he  allows  his  attorney  5%.  He  receives  a  further 
dividend  of  20%,  on  which  he  allows  his  attorney  5%.  What  is 
the  net  amount  that  he  receives  ? 

5.  If  a  commission  of  $212.94  is  paid  for  buying  $6552 
worth  of  goods,  find  the  rate  per  cent  of  commission. 

6.  An  agent  received  $40.62i  for  selling  a  house  for  $1625. 
Find  his  rate  per  cent  of  commission. 

7.  An  agent,  who  is  paid  a  commission  on  what  he  invests, 
received  $4896,  and  invests  $4800.  Find  his  rate  per  cent  of 
commission. 

8.  An  agent  received  $56  for  selling  grain  on  a  commission 
of  4%.     Find  value  of  grain  sold. 

9.  A  commission  merchant  charged  2i%  for  selling  a  quantity 
of  pork,  and  received  for  his  commission  $64.82.  Find  the  sell- 
ing price  of  the  pork. 

10.  The  owner  of  a  house  offered  an  agent  $500  commission, 
if  the  agent  could  sell  the  house  for  $10,500.  What  rate  per 
cent  commission  was  the  owner  offering  ?  Had  the  owner  offered 
5%  commission,  what  would  have  been  the  commission  on 
$10,500? 

11.  I  bought  a  bicycle  for  $70,  which  was  J  of  my  commission 
at  3^%  for  selling  a  quantity  of  land.  For  how  much  was  the 
land  sold  ? 

12.  A  real  estate  dealer  sold  land  for  100  units  of  money,  on 
a  commission  of  4%.  How  many  units  of  money  did  he  keep  for 
his  commission  ?  How  many  units  of  money  did  he  send  his 
employer?  If  his  employer  received  $2880,  what  did  the  land 
sell  for  ?     What  was  the  agent's  commission  ? 


INSURANCE  261 

13.  An  agent  remits  $4850  to  his  employer  after  taking  out 
his  commission  of  3%.     Find  the  selling  price. 

14.  My  agent  sent  me  as  my  share  of  the  selling  price  of  flour 
$2038.40.  If  the  flour  sold  for  $3.25  a  barrel,  and  the  agent's 
commission  was  2%,  how  many  barrels  did  he  sell  ? 

15.  My  agent  bought  a  quantity  of  goods  for  me  on  a  commis- 
sion of  2%.  If  the  cost  of  the  goods  was  100  units  of  money, 
how  many  units  of  money  did  his  commission  equal  ?  How  many 
units  did  I  have  to  send  him  to  cover  the  cost  of  the  goods  and 
his  commission  ? 

16.  A  merchant  in  Buffalo  sends  a  commission  merchant  in 
New  York  $3120,  instructing  him  to  purchase  goods,  reserving 
his  commission  at  4%.     Find  his  commission. 

17.  A  merchant  sent  $3238.30  to  New  Orleans  to  be  expended 
in  cotton.  The  broker  in  New  Orleans  charged  6%  commission. 
What  sum  was  paid  for  the  cotton  ? 

18.  Sent  to  a  commission  merchant  in  Chicago  $2080.80  to 
invest  in  flour,  his  commission  being  2%  on  the  amount  expended. 
How  many  barrels  of  flour  'could  be  purchased  at  $  4.25  a  barrel  ? 

19.  A  real  estate  broker  sold  a  house  on  3|^%  commission,  and 
sent  to  the  owner  $6176.  What  was  the  broker's  commission, 
and  what  sum  did  he  receive  for  the  house? 

20.  I  send  $5250  to  a  commission  merchant  in  St.  Louis,  who 
charges  5%  for  investing,  with  instructions  to  purchase  certain 
goods,  deducting  his  commission  from  the  amount  of  money  sent 
him.     Find  his  commission. 

INSURANCE 

203.  Insurance  is  a  contract  by  which  a  person  whose 
property  is  insured  receives  security  against  loss  by  fire  or 
accident  in  consideration  of  a  sum  of  money  paid  to  the 
insurance  company. 


262 


AHTTIIMETIC 


The  Premium  is  the  sum  paid  for  insurance.     It  is  always 
a  certain  per  cent  of  the  sum  insured. 

The  Policy  is  the  written  contract  of  insurance. 

204.   A  factory  valued  at  135,000  was  insured  for  f  of 

its  value,  the  rate  of  insurance  being  |^  for  one  year.    What 

was  the  amount  of  the  premium  ? 

The  premium  =  |%  of  f  of  $35,000. 

^JL^3^  $35000^      3125. 
800      5  1 


Exercise  166 

1.  A  store  worth  $4500  was  insured  for  f  its  value  at  1^%- 
Find  the  premium. 

2.  A  store  worth  $  4800  is  insured  for  J  of  its  value  at  $  1.55 
per  $  100.     Find  the  premium. 

3.  Find  the  premiums  paid  to  insure  property  against  loss  by 
fire  for  the  following  amounts,  at  the  given  rates : 

(a)  $  2500  at  $  1.15  per  $  100 ;  $  3600  at  $  1.20  per  f  100. 

(b)  $  8400  at  f  1.45  per  $  100 ;  $  7600  at  $  1.25  per  $  100. 

(c)  $  9600  at  $  1.60  per  $  100 ;  $  8500  at  $  1.50  per  $  100. 

4.  The  rate  of  insuring  an  hotel,  insured  for  $75,000,  was 
advanced  from  $  1.75  to  $  2.32  per  f  100.  Find  the  increase  m 
the  premium. 

5.  Find  the  increase  in  the  premiums  on  the  following  proper- 
ties, the  old  and  new  rates  and  the  amounts  of  insurance  being 
given : 


Amount  Insurkd 

Oi.o  Rate 

Nkw  Ratk 

Amount  Insurkd 

Old  Uatk 

Nkw   Katk 

$25,000 
90,000 
87,500 

$1.50 
1.26 
1.25 

$2.22 
1.81 
1.00 

$85,000 
45,000 
72,500 

$1.25 
1.40 
1.25 

$1.97 
1.60 
1.63 

INSURANCE  263 

6.  rind  the  entire  premium  for  insuring  a  house  for  3  yr.  for 
1 3000  at  $  1.75  per  $  100,  and  its  contents  for  $  1200  at  the  same 
rate. 

7.  A  house  is  worth  $4500  and  its  contents  $1800.  Find 
the  entire  premium  paid  for  insuring  both  at  |  of  their  value,  the 
rate  being  1|%. 

8.  A  fire  insurance  company  charged  $  262.50  for  insuring  a 
house  for  $  17,500.     What  was  the  rate  per  cent  of  insurance  ? 

9.  A  merchant's  stock  was  worth  $  120,000.  He  insured  it 
at  I  its  value,  paying  $  600  premium.  What  was  the  rate  per 
cent  of  insurance  ?     What  was  the  rate  in  cents  per  $  100  ? 

10.  A  shipment  of  goods  is  insured  for  $7500  at  j%.  Find 
the  premium. 

11.  For  what  sum  was  a  house  insured  if  the  premium  paid 
was  $  17.50  and  the  rate  of  insurance  J%  ? 

12.  For  what  sum  was  a  shop  insured  if  the  rate  of  insurance 
was  ()o  ^  per  $  100  and  the  premium  paid  was  $  81.25  ? 

13.  An  insurance  company  took  a  risk  at  2J%,  and  reinsured  f 
of  the  risk  at  2%.  The  premium  received  exceeded  the  premium 
paid  by  $  42.     Find  the  amount  of  the  risk. 

14.  What  will  be  the  cost  of  insuring  a  cargo  of  24,000  bu.  of 
wheat  valued  at  72  ^  per  bushel,  the  insurance  covering  J  of  the 
value  of  the  cargo,  the  premium  rate  being  1|%  ? 

15.  A  merchant's  stock  was  insured  for  $42,000,  i  of  this 
amount  being  at  |%,  |  of  the  remainder  at  f%,  and  the 
remainder  at  |%.     Find  the  total  amount  of  premium  paid. 

16.  A  merchant  insured  his  stock  for  $33,000  for  one  year 
at  1^%.  Six  months  thereafter  the  policy  was  cancelled  at  the 
request  of  the  insured.  Find  the  amount  of  premium  returned, 
the  short  rate  for  six  months  being  |%. 

17.  A  warehouse  valued  at  $62,500  was  insured  for  f  of  its 
value.  The  rate  of  insurance  was  1J%.  What  was  the  amount 
paid  for  the  insurance  ? 


264  ARITHMETIC 

18.  A  factory  and  the  machinery  therein  is  insured  for 
$65,000;  f  of  this  sum  is  at  |%  premium  and  the  remainder 
is  at  J%.     What  is  the  entire  premium? 

19.  A  fire  insurance  company  received  $350  for  insuring  a 
factory  at  1^%  premium.     What  was  the  amount  of  insurance? 

20.  A  building  and  contents  are  insured  as  follows:  $12,000 
in  the  first,  $  8000  in  the  second,  and  $  5000  in  the  third  insurance 
company.  Were  a  loss  to  the  extent  of  $  3500  to  occur  through 
fire,  what  portion  of  the  loss  should  each  company  bear  ? 

21.  Merchandise  valued  at  $63,000  was  insured  in  the  first 
insurance  company  for  $  15,000,  in  the  second  for.  $  12,000,  and 
in  the  third  for  $  8000.  If  the  merchandise  is  damaged  by  fire 
to  the  extent  of  $  10,500,  how  much  of  the  damage  should  each 
company  pay  ? 

22.  A  steamboat  worth  $  60,000  is  insured  in  three  companies ; 
in  two  to  the  amount  of  $  15,000  each,  and  in  the  third  to  the 
amount  of  $  20,000.  For  what  sum  would  each  company  be  liable 
if  the  vessel  were  to  sustain  damage  to  the  extent  of  $  6600  ? 

TAXES  AND  DUTIES 

205.  A  Tax  is  a  sum  of  money  assessed  on  persons  or 
property  for  public  purposes. 

The  tax  on  property  is  reckoned  at  a  certain  rate  per  cent 
of  the  assessed  value  of  the  property. 

Direct  taxes  are  levied  by  the  state,  county,  township,  city, 
or  the  school  district. 

Some  states  levy  a  tax  upon  each  voter,  independent  of 
the  property  he  owns.  Such  a  tax  is  called  a  Poll-tax,  and 
as  a  rule  does  not  exceed  $  2  a  year. 

Indirect  taxes,  called  Duties,  are  levied  by  the  general  gov- 
ernment on  imported  goods. 

An  Ad  Valorem  Duty  is  reckoned  at  a  certain  rate  per  cent 


TAXES  AND  DUTIES  265 

of  the  cost  of.  the  goods  in  the  country  from  which  they  have 
been  imported. 

A  Specific  Duty  is  a  fixed  charge  on  the  quantity  of  goods 
without  reference  to  their  cost,  as  a  specific  tax  of  one  cent 
a  pound. 

206.  The  people  of  a  school  section  wish  to  build  a  new 
schoolhouse,  which  will  cost  $  2850.  The  taxable  property 
of  the  section  is  assessed  at  $  190,000 ;  what  will  be  the  rate 
of  taxation,  and  what  will  be  the  tax  on  property  assessed  at 

17500? 

The  tax  on  $  190,000  =  $  2850. 
.-.  the  rate  of  taxation  =  $2850  ^  $  190,000  =  .015  or  1^%. 
.-.  the  tax  on  $  7500  =  1^ %  of  $  7500  =  $  112.50. 

Exercise  167 

1.  State  expenses  which  the  government  meets  by  taxation; 
the  county ;  the  township ;  the  village  or  city.;  the  school  district. 

2.  A  village  levies  a  tax  of  $  12,000  and  pays  the  tax  collector 
2%  and  $150  for  collecting  the  taxes.  Find  the  net  amount  of 
the  taxes. 

3.  A  village  levied  a  tax  of  $  15,000  and  paid  the  tax  collector 
2%  and  $200  for  collecting  the  taxes.  If  f  250  taxes  were  non- 
collectible,  find  the  net  amount  of  the  taxes. 

4.  What  is  the  tax  on  property  assessed  at  $  6400,  the  rate  of 
taxation  being  1^%  ? 

5.  Find  the  tax  on : 

'      (a)  $8000  at  6  mills  on  a  dollar. 

(6)  $4500  at  8  mills  on  a  dollar. 

(c)  $2800  at  51-  mills  on  a  dollar. 

(d)  $4800  at  9.2  mills  on  a  dollar. 

(e)  $3750  at  6.4  mills  on  a  dollar. 
(/)  $7250  at  8.6  mills  on  a  dollar. 


266  ARITHMETIC 

6.  A  man  owns  a  farm  of  80  A.,  worth  $  60  an  acre.  What  is 
his  tax  at  8  mills  on  a  dollar,  his  property  being  assessed  at  J  of 
its  value  ? 

7.  A  man  whose  property  is  assessed  for  ^  7500  pays  the  fol- 
lowing taxes :  state  1^  mills,  town  .6  mill,  and  school  1^  mills. 
Find  his  total  tax. 

8.  In  a  school  section  a  tax  of  $  4000  is  to  be  raised.  If  the 
assessed  valuation  of  the  property  is  $250,000,  what  will  be  the 
tax  on  the  dollar,  and  what  is  A's  tax,  whose  property  is  valued 
at  $  1800  ? 

9.  What  is  the  assessed  value  of  property  taxed  $  37.50,  at 
the  rate  of  15  mills  on  the  dollar  ? 

10.  What  is  the  assessed  value  of  property  taxed  $37.80,  at 
the  rate  of  4|-  mills  on  the  dollar  ? 

11.  A  man  whose  property  is  taxed  16  mills  on  a  dollar  pays 
a  tax  of  $  134.40.     Find  the  assessed  value  of  his  property. 

12.  A  village  makes  the  following  tax  levy  at  the  rate  of  3.4 
mills  on  a  dollar:  police  $2200,  roads  $2500,  trea.surer  $600, 
attorney  $200,  clerk  $300,  tax  collector  $300,  miscellaneous 
$  2400.     Find  the  assessed  value  of  the  property  in  the  village. 

13.  The  municipal  rates  being  reduced  from  19|  mills  to  17 J 
mills  on  the  dollar,  my  taxes  are  lowered  by  $4.05.  For  how 
much  am  I  assessed  ? 

14.  In  a  certain  section  a  schoolhouse  is  to  be  built  at  an 
expense  of  $  8400,  to  be  defrayed  by  a  tax  upon  property  valued 
at  $  700,000.  What  is  the  rate  of  taxation  to  cover  both  the  cost 
of  the  schoolhouse  and  the  collector's  commission  of  $  350  ? 

15.  The  assessed  valuation  of  a  town  is  $972,250,  and  the 
town  has  320  polls  paying  $1.50  each;  what  is  the  rate  of  taxa- 
tion when  the  tax  levy  is  $19,025?  What  tax  must  a  person 
pay  whose  property  is  assessed  for  $  7500,  and  who  pays  for  one 
poll? 


MISCELLANEOUS  EXERCISE  267 

16.  If  the  assessed  value  of  a  town  is  $1,260,000,  and  the 
town  has  420  polls  paying  $  1.25  each,  what  is  the  rate  of  taxa- 
tion on  property  when  the  tax  levy  is  f  15,645  ?  What  does  A 
pay,  whose  property  is  assessed  at  $  8500  and  who  pays  one  poll  ? 

17.  A  house  assessed  at  $2200  was  rented  for  $23  a  month, 
the  tenant  to  pay  taxes  and  water-rates.  The  taxes  were  17 
mills  on  the  dollar,  and  the  water-rates  were  $  5  per  quarter  year. 
How  much  all  together  did  the  tenant  pay  per  year  for  the  house  ? 
If  the  property  had  cost  the  landlord  $  2500,  what  rate  per  cent 
per  year  was  he  receiving  on  his  investment  ? 

18.  What  is  the  duty  on  600  boxes  of  cigars  costing  $  7.50  a 
box,  the  duty  on  cigars  being  35  %  ? 

19.  What  is  the  duty  on  800  yd.  cloth  at  12s.  a  yard,  the  duty 
being  25%?     (1  £  =  $4.86.) 

20.  What  is  the  duty  on  600  yd.  of  cloth  invoiced  at  6  francs 
per  yard,  the  duty  being  30%  ?     (1  franc  =  19.3^.) 

Miscellaneous  Exercise  168 

1.  I  bought  an  article  for  $  3.60  and  sold  it  for  $  4.20.    What 
is  my  gain  per  cent  ? 

2.  I  sold  goods  for  $  3360  and  gained  12  % .    What  was  the  cost 
price  ? 

3.  If  425  yd.  of  silk  are  sold  for  $  1657.50,  and  20%  profit  is 
made,  what  did  it  cost  per  yard  ? 

4.  By  selling  goods  for' $1088,  I  lost  16%.     How  much  per 
cent  should  1  have  lost  or  gained  if  I  had  sold  them  for  $  1344  ? 

5.  A  tradesman's  prices  are  25%   above  cost  price.     If  he 
allows  a  customer  8%  on  his  bill,  what  profit  does  he  make  ? 

6.  8%   is  gained  by  selling  a  piece  of  ground  for  $8251.20; 
what  would  be  gained  per  cent  by  selling  it  for  $  8404  ? 

7.  Find  the  brokerage  on  $  1324  at  i%. 

8.  Find  the  brokerage  on  $  375  at  5%. 


268  ARITHMETIC 

9.   What  amount  of  money  was  invested,  when  the  broker's 
charges  at  1^%  amounted  to  $150? 

10.  My  agent  has  purchased  real  estate,  on  my  account,  to  the 
amount  of  $19,384.     What  is  his  commission  at  1^%  ? 

11.  The  price  of  flour  per  barrel  at  different  times  during  the 
year  1900  was : 

June  18 .'54.20 

June  19 '.     .     .      4.40 

June  21 4.65 


April  26 $3.80 

April  30 3.90 

June  8 4.00 

June  9 4.10 


June  22 4.76 


Find  the  rate  per  cent,  at  each  date,  above  or  below  the  price  on 
June  8. 

12.  The  price  of  cotton  advanced  from  6.3  ^  to  9.3  ^  per  pound. 
Find  rate  per  cent  increase  in  price. 

13.  What  is  the  premium  of  insurance  on  the  contents,  insured 
for  $1500  at  2^%? 

14.  What  is  the  premium  for  insuring  a  cargo  for  $  16,450,  at 
3i%? 

15.  A  person  at  the  age  of  40  insures  his  life  in  each  of  two 
offices  for  $4500,  the  premiums  being  at  the  rate  of  3 J  and  3|% 
respectively.     Find  his  annual  payment. 

16.  A  trader  gets  600  bbl.  of  flour  insured  for  80%  of  its  cost, 
at  2^%,  paying  $37.80  premium.  At  what  price  per  barrel  did 
he  purchase  the  flour  ? 

17.  A  shipment  of  dry  goods  was  insured  at  1|%  to  cover  ^  of 
its  value.    The  premium  was  $  28.    What  were  the  goods  worth  ? 

18.  A  man  who  owns  $  12,750  worth  of  property  pays  a  tax  of 
$  216.75.     Find  the  rate  on  the  dollar. 

19.  A  certain  town  has  property  assessed  at  $520,000,  and 
levies  a  tax  of  $  7800.  What  should  B  pay,  whose  property  is 
assessed  at  $  2500  ? 

20.  A  town  has  levied  a  tax  of  $  7.690,  which  sum  includes  the 
amount  voted  for  building  a  town  hall  and  the  collector's  fees,  at 
3%.    What  was  expended  on  the  town  hall  ? 


MISCELLANEOUS   EXERCISE  269 

21.  What  is  the  rate  per  cent  of  commission  when  I  receive 
$  5  for  selling  goods  to  the  value  of  $  125  ? 

22.  I  sold  a  quantity  of  goods  for  $  273.68,  on  a  commission  of 
2f  %.     Find  my  commission. 

23.  A  and  B  insure  their  houses  against  fire,  and  A  has  to  pay 
17.50  more  than  B,  who  pays  $28.75.  Find  the  amount  for 
which  their  houses  are  insured,  the  rate  of  insurance  being  |-%. 

24.  A  merchant  bought  goods  amounting  to  $  7460  subject  to 
25  and  5  off,  f  3730  subject  to  30  off,  and  $  1492  subject  to  20 
and  10  off.     Find  the  net  cost  of  the  goods, 

25.  If,  in  the  preceding  example,  the  invoice  clerk  were  to  bill 
all  the  goods  subject  to  30  off,  what  would  be  the  error  in  their 
net  cost  ? 

26.  A  man  bought  a  house  and  lot  for  $  2250.  After  spending 
$  630  on  repairs  and  improvements,  and  paying  $  30  for  taxes 
and  other  expenses,  he  sold  the  property  for  $  3880.  What  rate 
per  cent  of  profit  did  his  investment  yield  him  ? 

27.  In  an  examination  A  obtained  78%  of  the  full  number  of 
marks,  beating  B  by  16%  of  the  full  number.  If  A  received  975 
marks,  how  many  did  B  receive  ? 

28.  By  selling  a  certain  book  for  $  3.96,  I  would  lose  12%  of 
the  cost.  What  advance  on  this  proposed  selling  price  would 
give  a  profit  of  12%  of  the  cost? 

29.  Goods  are  sold  at  a  loss  of  20%  on  the  cost.  By  what 
percentage  of  itself  should  the  selling  price  be  advanced  to  yield 
a  profit  of  20%  on  the  cost  ? 

30.  A  man  having  bought  a  certain  quantity  of  goods  for  $  150, 
sells  ^  of  them  at  a  loss  of  4%.  By  what  increase  per  cent  must 
he  raise  that  selling  price  that  by  selling  the  whole  at  that  in- 
creased rate  he  may  gain  4%  on  his  entire  outlay? 

31.  The  cost  of  freight  and  insurance  on  a  certain  quantity  of 
goods  was  15%,  and  that  of  duty  10%  on  the  original  outlay. 


270  ARITHMETIC 

The  goods  were  sold  at  a  loss  »of  5%,  but  had  they  brought  $S 
more  there  would  have  been  a  gain  of  1%.  How  much  did  they 
cost? 

32.  A  man  began  business  with  a  certain  capital;  he  gained 
20%  the  first  year,  which  he  added  to  his  capital,  and  37^%  the 
second  year,  which  he  added  to  his  capital ;  in  the  third  year  he 
lost  40%  ;  had  he  received  $  600  more  for  the  goods  sold  the  last 
year,  he  would  have  cleared  in  the  three  years  2%  of  his  original 
capital.     Find  the  capital  with  which  he  commenced  business. 

33.  A  merchant  bought  400  lb.  of  tea  and  1600  lb.  of  coffee, 
the  cost  of  the  latter  per  pound  being  16|%  that  of  the  former; 
he  sold  the  tea  at  a  profit  of  33|%,  and  the  coffee  at  a  loss  of 
20%,  gaining,  however,  on  the  whole  $60.  Find  his  buying 
prices  and  his  selling  prices. 

34.  A  sells  potatoes  for  $  1  per  bushel  and  gains  25%.  After- 
ward he  sold  some  of  the  same  lot  of  potatoes  to  the  amount  of 
$  36  and  gained  50%.  How  many  bushels  were  there  in  the  last 
lot  and  at  what  rate  per  bushel  did  he  sell  them  ? 

35.  A  person  marks  his  goods  so  that  he  may  allow  a  discount 
of  4%,  and  still  make  a  profit  of  15%.  What  must  be  the  marked 
price  of  an  article  that  cost  him  f  4.80  ? 

36.  A  manufacturer  who  employed  men  at  $  1.60  a  day  found 
that  he  could  save  15%  by  employing  women.  What  wages  were 
paid  the  latter,  supposing  a  woman  could  do  J  as  much  as  a  man 
in  the  same  time  ? 

37.  A  merchant  buys  goods;  the  cost  of  freight  is  8%,  and 
that  of  insurance  12%  on  the  original  outlay;  he  is  obliged  to 
sell  them  at  a  loss  of  7%  ;  but  if  he  had  received  $  5.10  more  for 
them  he  would  have  gained  1^%.     Find  the  original  outlay. 

38.  A  merchant  sells  50  yd.  of  broadcloth  at  a  gain  of  15%, 
and  75  yd.,  which  cost  the  same  per  yard,  at  a  gain  of  10%,  and 
finds  that  if  he  had  sold  the  whole  at  a  uniform  i;ain  of  12.J-%,  he 
would  have  received  $2.25  more  than  he  actually  did  receive. 
What  was  the  cost  price  per  yard  ? 


MISCELLANEOUS  EXERCISE  271 

39.  A  man  buys  goods  for  a  certain  sum,  and  marks  ^  of 
them  at  a  profit  of  24%,  and  |  of  them  at  a  profit  of  36%  ; 
but  had  he  marked  ^  of  them  at  24%  gain,  and  i  at  36%  gain, 
he  would  have  realized  $  240  less  than  before.  Find  the  cost  of 
the  goods. 

40.  A  wheat  buyer  sold  ^  of  his  wheat  at  a  certain  gain  per 
cent,  i  of  it  at  a  gain  of  twice  the  former  rate  per  cent,  and  the 
remainder  at  a  gain  per  cent  of  3  times  the  first  gain.  If  the 
gain  on  the  entire  stock  was  26%,  what  did  he  gain  on  each  part  ? 
If  he  gained  5%  on  the  first  part,  what  was  the  entire  gain  per 
cent? 

41.  A  merchant  wishes  to  mark  some  goods  which  cost  $  1.20 
per  yard,  so  that  after  making  a  reduction  of  20%  off  the  marked 
prices,  he  may  yet  gain  10%.  At  what  price  per  yard  must  he 
mark  the  goods  ? 

42.  Sold  goods  to  a  certain  amount  on  a  commission  of  5%, 
and  having  remitted  the  net  proceeds  to  the  owner,  received  for 
prompt  payment  |^%,  which  amounted  to  $24,221.  What  was 
the  selling  price  of  the  goods  ? 

43.  What  single  discount  is  equivalent  to  successive  discounts 
of  20%  and  10%? 

44.  Show  that  successive  discounts  of  specified  rates  may  be 
taken  off  a  list  price  in  any  order  without  affecting  the  net  price. 
Thus  20  and  10  off  is  equivalent  to  10  and  20  off,  so  also  30  and 
10  and  5  off,  10  and  30  and  5  off,  and  5  and  30  and  10  off  are  all 
equivalent. 

45.  An  agent  sold  6  mowing-machines  at  $  120  each,  and  12  at 
$  140  each.  After  deducting  his  commission  he  remitted  $  2280 
to  his  employer.     What  was  the  rate  of  commission  ? 


CHAPTER  XV 

INTEREST 

207.  Interest  is  money  paid  for  the  use  of  money. 
The  Principal  is  the  sum  loaned. 

The  Amount  is  the  sum  of  the  principal  and  interest. 
The  Rate  of  Interest  is  always  expressed  as  a  rate  per  cent 
of  the  principal. 

The  unit  of  time  is  1  year. 

208.  (1)  What  is  the  interest  on  1638  for  1  yr.  at  6^? 

$638  principal 
.06  rate  per  unit 


$38.28  interest  for  1  yr. 
.*.  the  interest  on  $638  for  1  yr.  at  6%  =  6 %  of  $638  =  $38.28. 

(2)  Find  the  interest  on  and  the  amount  of  1473.28  for 
81  da.  at  7%. 

$473.28  principal 

.07  rate 
$33.1296  interest  for  1  yr. 

$33.13  interest  for  1  yr. 
9 


40)$298.17 

7.46  interest  for  81  da. 
_473.28 
$480.73  amount 

The  interest  for  1  yr.  =  7%  of  $473.28  =  $33.13. 

The  interest  for  81  da.  =  ^Vs  or  ^  of  $33.13  =  $7.45. 

The  amount  =  $473.28  +  7.46  =  $480.73. 
272 


INTEREST  273 

(3)  Find  the  amount  of  1385.35,  from  July  7,  1895,  to 
Oct.  13,  1895,  at  l^f. 

mo.         da. 

Oct.  13  =  10        13 

July  7=_7 7 

The  time  =3  6  =  3|  mo.  =  /^  yr. 

Therate  =  x\xi#%  =  2%. 
The  interest  =  2  %  of  $  385.35  =  $  6. 71. 
The  amount  =  $385.35  +  $6.71  =  $392.06. 

209.   Six  Per  Cent  Method. 

The  interest  at  6%  for  1  yr.  =  .06  of  the  principal. 

The  interest  at  6%  for  1  mo.  =  ^^  of  .06  or  .005  of  the  principal. 

The  interest  at  6%  for  1  da.  =  ^^  of  -005  or  .000^  of  the  principal. 

Find  the  interest  on  $435  for  9  mo. '24  da.  at  6^. 

The  interest  for  9  mo.  =    9  x  .005    =  .045 

The  interest  for  24  da.  ="24  x  .000^  =  .004 

The  interest  for  9  mo.  24  da.  =  .049 

.-.  the  interest  =  .049  x  $435  =  $21,315. 

To  find  the  interest  at  any  other  rate  per  cent,  divide  the 
interest  at  6%  by  6  and  multiply  by  the  given  rate  per  cent. 

The  interest  at  7|%  may  be  found  by  increasing  the  interest  at  6%  by 
-a  or  -;  that  at  b\%  by  diminishing  the  interest  at  Q%  by  2  or  — 

By  what  fraction  must  the  interest  at  6%  be  increased  in  order  to  give 
the  interest  at  each  of  the  following  rates  :   7%,  8%,  9%,  6^%,  6.2%  ? 

By  what  fraction  must  the  interest  at  6%  be  diminished  in  order  to  give 
the  interest  at  each  of  the  following  rates  :  6%,  4%,  3%,  4^%,  4:\X,  5|%  ? 

Exercise  169 
Find  the  interest  on : 

1.  $449  for  1  yr.  at  5%.  5.    $587.50  for  5  mo.  at  6%. 

2.  $757  for  1  yr.  at  4%.  6.    $628.90  for  9  mo.  at  4i%. 

3.  $643.17  for  1  yr.  at  7%.       7.    $323.75  for  60  da.  at8%. 

'  4.    $  725  for  4  mo.  at  8%.  8.    $ 958.50  for  90  da.  at  4i%. 


274  ARITHMETIC 

9.    f28G5for33da.  atG%.       12.    ^225.90  for  03  da.  at  7%. 

10.  $312.80for  93  da.  atG%.     13.    ^390.50  for  93  da.  at  6%. 

11.  $612.94for33da.at7i%.     14.    $8396.40forl23da.at8%. 

15.  1^4087.50  fori  mo.  3  da.  at  9%, 

16.  $  1465.53  for  3  mo.  3  da.  at  5%. 

17.  $  1350  for  3  mo.  21  da.  at  7%. 

18.  ^295.36  for  57  da.  at  6.2%. 

19.  f  1200  from  May  7  to  June  6  at  7%. 

20.  f  975.65  from  Sept.  16  to  Dec.  8  at  6J%. 

21.  $450  from  Sei3t,  4  to  Oct.  27  at  7%. 

22.  .^79.50  from  Deo.  23  to  Feb.  20  of  the  next  year  at  7^%. 

23.  f  586.67  from  Jan.  15,  1901,  to  May  1,  1901,  at  8%. 

24.  State  how  to  find  the  simple  interest  when  the  principal, 
rate  per  cent,  and  time  are  given. 

25.  Name  the  terms  in  problems  in  Profit  and  Loss,  Commis- 
sion and  Insurance,  which  correspond  to  principal  and  rate  per 
cent  of  interest. 

26.  Find  the  relation  between  the  interest,  principal,  and 
amount,  when  the  time  is  3  mo.,  and  rate  8%;  time  120  da., 
rate  9%. 

27.  Find  the  amount  of  $473.28  for  3  mo.  at  ^%  per  month. 

28.  Find  the  amount  of  $885.85  for  1  mo.  15  da.  at  5%. 

29.  Find  the  amount  of  $628.25  for  185  da.  at  41%. 

30.  Find  the  amount  of  $935.68  for  66  da.  at  6^%. 

31.  Find  the  amount  of  $  147.50  for  93  da.  at  7%. 

32.  Find  the  amount  of  $250  from  July  9  to  Aug.  18  at  8%. 

33.  Find  the  amount  of  $2394  from  May  8  to  Sept.  21  at  4%. 

34.  Find  the  amount  of  $5246  from  March  1  to  Au^.  3  at  5%. 

35.  Find  the  amount  of  $  230.80  from  Jan.  4,  1901,  to  June  23, 
1901,  at  6%. 

36.  Find  the  amount  of  $657.60  from  Aug.  9  to  Deo.  5  at  8%. 


INTEREST  275 

37.  A  person  loaned  $  480  for  2  mo.  and  13  da.  at  9  %.  What 
interest  did  he  receive  ? 

38.  On  March  20,  a  merchant  sold  goods  to  the  value  of  $  1168, 
and  received  a  note,  due  June  8,  next,  for  that  sum  with  interest 
at  7  %  per  annum.     For  what  amount  was  the  note  drawn  ? 

39.  A  debt  of  $  175  became  due  on  June  13,  after  which  date 
interest  was  charged  at  the  rate  of  8  % .  What  must  be  paid  to 
settle  the  debt  Sept.  14  ? 

40.  A  owes  $15,000  bearing  interest  at  5%  per  annum;  he 
pays  at  the  end  of  each  year  for  interest  and  part  payment  of 
principal  $  2500.  Find  the  amount  of  his  debt  at  the  end  of  the 
third  year. 

41.  A  man  engaged  in  business  with  a  capital  of  $  10,920  is 
making  12i%  per  annum  on  his  capital,  but  on  account  of  ill 
health  he  quits  the  business  and  loans  his  money  at  5%.  How 
much  is  his  income  diminished  ? 

42.  $  420.  Chicago,  June  4,  1901. 
Sixty  days  from  date  I  promise  to  pay  Samuel  Jones,  or  order, 

four  hundred  and  twenty  dollars,  with  interest  at  six  per  cent, 
value  received.  Eichaed  Walsh. 

What  is  the  amount  of  this  note  at  maturity  ? 

43.  A  merchant  borrows  $1600  for  1  yr.  at  7%.  Find  what 
he  owes  at  the  end  of  the  year.  In  case  he  pays  only  $  12  inter- 
est, how  much  will  he  owe  at  the  beginning  of  the  next  year  ? 
What  will  he  owe  at  the  end  of  the  year  ? 

44.  If  I  borrowed  $1200  Jan.  1,  1900,  at  6%,  what  would  I 
owe  Jan.  1,  1901  ?  If  I  kept  the  money  until  Jan.  1,  1902,  what 
would  I  then  owe  ? 

EXACT   INTEREST 

210.  In  order  to  find  the  exact  interest  we  must  reckon 
365  da.  to  a  year.  Exact  interest  is  used  by  the  United 
States  Government  and  sometimes  in  business  transactions. 


276  ARITHMETIC 

211.  The  exact  interest  at  5  %  for  1  da.  is  yf 3,  or  ^^  of  the  principal. 
The  common  interest  is  ^f  j,  or  ^^  of  the  principal.     Therefore  the  exact 

interest  is  ^^^  -r-  7^  or  f|  of  the  common  interest.     Hence  the  exact  interest  is 
equal  to  the  common  interest  diminished  by  7^  of  itself. 

212.  Find  the  exact  interest  on  14250  from  May  12  to 
Oct.  3  at  7%. 

The  number  of  days  from  May  12  to  Oct.  3  =  19  -}-  30  +  31  +  31  +  30  + 
3  =  144. 

The  interest  on  $4250  at  7  %  for  1  yr.  =  $297.60. 

The  interest  on  $4250  at  7  %  for  144  da.  =  ||f  of  $297.50  =  $  117.37. 

Exercise  170 
Find  the  exact  interest  on : 

■     1.    $  2450  for  146  da.  at  6%. 

2.  $  3475  for  292  da.  at  7%. 

3.  $  1560  for  60  da.  at  5%. 

4.  ^  629  for  113  da.  at  6%. 

5.  $  1400  from  July  6  to  Dec.  4  at  5J%. 

6.  $  1850  from  March  1  to  Aug.  6  at  6^%. 

7.  $  2500  from  May  1  to  Sept.  24  at  5%. 

8.  $  2480  from  Aug.  9  to  Sept.  18  at  6%. 

BANK  DISCOUNT 

213.  A  merchant,  who  desires  to  obtain  a  loan  of  fSOO 
for  90  da.,  makes  a  note  and  takes  it  to  the  bank,  which 
deducts  the  interest  on  $  800  for  90  da.  at  a  certain  rate  per 
cent,  which  varies  from  time  to  time.  This  bank  gives  him 
the  proceeds^  and  collects  the  $  800  at  the  end  of  90  da. 

In  those  states  that  have  not  abolished  days  of  grace, 
three  days,  known  as  dai/s  of  grace^  are  added  to  the  specified 
time  to  find  when  the  payment  is  due.  In  these  states  the 
bank  discount  in  the  above  instance  would  be  reckoned  for 


BANK  DISCOUNT 


277 


93  da.  The  note  would  then  be 
grace  have  been  abolished  in  all 
(Jan.,  1901)  : 

Alabama 

Arizona  Territory 
Arkansas 


Georgia 

Indiana 

Indian  Territory 

Iowa 

Kansas 


Kentucky 

Louisiana 

Michigan 

Minnesota 

Mississippi 

Missouri 

Nebraska 

Nevada 


due  in  93  da.     Days  of 
but  the  following  states 

New  Mexico  Territory 
North  Carolina 
Oklahoma  Territory 
South  Carolina 
South  Dakota 
Texas 
Wyoming 
Canada 


Make  a  list  of  those  states  that  have  abolished  days  of 
grace. 

214.  Bank  Discount  is,  therefore,  simple  interest  collected 
in  advance  upon  the  sum  due  on  a  note  at  its  maturity. 

Nearly  all  notes  specify  the  place  of  payment.  In  case  the 
place  of  payment  is  not  specified  in  the  note,  it  is  to  be  paid 
at  the  business  office  of  the  maker  of  the  note. 

215.  f  450. 75.  Chicago,  July  3,  1901. 
Sixty  days  after  date  I  promise  to  pay  to  the  order  of  James 

Smith,  four  hundred  fifty  and  -^^^  dollars  at  the  First  National 
Bank.     Value  received.  Horace  Ward. 

Discounted  July  3,  at  6  %.     Find  proceeds. 

The  discount  =  the  interest  on  $450.75  at  6%  for  60  da.  =  $4.51. 
The  proceeds  =  $450.75  -  $4.51  =  $446.24. 

In  those  states  that  have  not  abolished  days  of  grace  use  63  da.  The  dis- 
count will  then  be  $4.78  and  the  proceeds  $446.02. 

216.  The  Day  of  Maturity  is  the  day  on  which  the  note 
becomes  legally  due. 

The  Proceeds  of  a  Note  is  the  sum  of  money  received  for  it 
when  discounted. 


278  ARITHMETIC 

It  is  found  by  subtracting  the  discount  from  the  value  of 
the  note  at  maturity. 

The  Time  to  run  is  the  time  between  the  day  on  which  the 
note  is  discounted  and  the  day  of  maturity. 


Exercise  171 

1.  $  600.  Chicago,  July  6,  1898. 

Thirty  days  after  date  I  promise  to  pay  to  George  Boies,  or 
order,  six  hundred  dollars,  value  received.         Robert  Brown. 

Discounted  at  7  %,  July  6,  1898.     Find  proceeds. 

Face  of  Note  Date  of  Note  Time     Rate  of  Discount 

2.  $  312.80;      May  13,  1899 ;  90  da. ;      6  %.      Find  proceeds. 

3.  $  225.90 ;      June  14,  1896 ;     2  mo. ;     7  % .      Find  proceeds. 

4.  f  100.00 ;      Feb.  12,  1898 ;  30  da. ;      5  %.      Find  proceeds. 

5.  State  how  to  find  the  proceeds  of  any  note  discounted  at 
once. 

6.  Write  the  notes  corresponding  to  examples  2  and  3. 

7.  $  390 i%V  Springfield,  III.,  May  1,  1900. 

Three  months  after  date   I   promise  to  pay  to  the  order  of 
Thomas   A.  Stuart,  three  hundred  ninety  and  j^^^  dollars.     Value 

^®^®^^^^-  James  Henderson. 

Discounted  May  1,  1900,  at  6%.    Find  proceeds. 

217.    (1)  $  712.65.  .       Chicago,  July  6,  1899. 

Sixty  days  from  date  I  promise  to  pay  George  Wilson, 
or  order,  seven  hundred  twelve  and  -^  dollars,  for  value 

^^°'^^^^'^-  Samuel  Jones. 

Discounted  at  7%,  Aug.  6,  1899. 


BANK  DISCOUNT  279 

In  the  above  note,  find  the  day  of  maturity^  the  time  to  run, 
the  discount,  and  the  proceeds. 

The  day  of  maturity   =  60  da.  after  July  6  =  Sept.  4,  1899. 

The  time  to  run  =  the  number  of  days  between  Aug,  6  and  Sept.  4. 

=  29  da.  =  ^\\  yr. 
The  discount  =  the  interest  on  $  712.65  for  29  da.  at  7  %  =  $  4.02. 

Tlie  proceeds  =  $  712.65  -  $  4.02  =  $  708.63. 

In  those  states  that  have  not  abolished  days  of  grace  the  results  are : 
Sept.  7,  32  da.,  $4.43,  $708.22. 

(2)  I  450.76.  New  York,  May  5,  1901. 

Three  months  after  date,  for  value  received,  I  promise  to 
pay  Thomas  King,  or  order,  four  hundred  fifty  and  -^^^  dollars, 
at  the  First  National  Bank,  with  interest  at  6  fo. 

Arthur  Hill. 

Discounted  July  1,  1901,  at  8%. 

The  day  of  maturity  =  3  mo.  after  May  5  =  Aug.  5,  1901. 

The  amount  of  the  note,  Aug.  5,  1901  =  the  amount  of  $450.76  for  3  mo. 

at  6%  =  $467.52. 
The  time  to  run  =  the  number  of  days  between  July  1  and  Aug.  5  =  35  da. 
The  Discount     =  the  interest  on  $ 457.52  for  35  da.  at  S%  =  $ 3.56. 
The  proceeds      =  $457.52  -  $3.56  =  $453.96. 

In  those  states  that  have  not  abolished  days  of  grace  the  results  are  : 
Aug.  8,  $457.75,  38  da.,  $3.87,  $453.88. 

In  the  following  exercise  find  the  day  of  maturity,  the  time 
to  run,  the  discount,  and  the  proceeds.  If  the  state  in  which 
you  live  has  not  abolished  days  of  grace  add  them,  otherwise, 
not. 

Exercise  172 

1.    $  2400.  Cleveland,  O.,  March  3,  1901. 

Three  months  after  date  I  promise  to  pay  Ralph  Barker,  or 
order,  twenty-four  hundred  dollars,  value  received. 

EoBERT  Peterson. 

Discounted  at  7  %,  May  7. 


280  ARITHMETIC 

2.  A  note  for  $572.80,  drawn  on  June  13  and  payable  4  mo. 
after  date,  was  discounted  at  7%  on  June  27. 

3.  $  2400.  Cleveland,  0.,  March  3,  1898. 
Three  months  after  date  I  promise  to  pay  Ralph  Barker,  or 

order,  twenty -four  hundred  dollars,  for  value  received,  with  in- 
terest at  6%.  KoBERT  Peterson. 
Discounted  at  7  %,  May  7. 

4.  State  business  transactions  which  may  have  preceded  the 
giving  of  the  notes  in  examples  1,  2,  and  3. 

5.  On  July  7,  James  Monroe  bought  a  farm  from  John  Harris, 
paying  $  2000  cash  and  giving  his  note,  without  interest,  for  $  1200, 
payable  in  60  da.     Write  the  note. 

Face  of  Note  Date  of  Note  Time  Date  of  Disc.  Rate  of  Disc. 

6.  $312.80;  May  13,1899;  90  da. ;    May   13;  6^%. 

7.  $975.65;  Sept.    5,1899;     3  mo. ;  Sept.  16 

8.  $450.00;  Aug.  28,1901;  60  da. ;    Sept.    4: 

9.  $79.50;  Dec.  17,1899;     2  mo. ;  Dec.   23 

10.  $586.67;  Dec.   28,1901;     4  mo. ;  Jan.   15,1902;  8%. 

11.  $2480.  Buffalo,  N.Y.,  Nov.  19,  1900. 
Six  months  after  date  I  promise  to  pay  Alfred  Jameson,  or 

order,   two   thousand   four    hundred    and   eighty   dollars,   value 
received,  with  interest  at  5%.  William  O'Connor. 

Discounted  at  6  %,  Jan.  4,  1901. 

12.  State  how  to  find  the  proceeds  of  a  note,  not  bearing  inter- 
est, when  discounted.  What  change  is  to  be  made  in  the  solution 
when  the  note  bears  interest  ? 

13.  $  2065.76.  New  Orleans,  June  4,  1899. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of  Edgar 

Johnston  two  thousand  sixty-five  and  -^^  dollars,  for  value  re- 
ceived, with  interest  at  6%.  Alexander  Grant. 
Discounted  at  8%,  July  4,  1899. 


7%. 


COMPOUND  INTEREST  281 

COMPOUND  INTEREST 

218.  Compound  Interest  is  interest  which  is  found  for  stated 
periods  and  added  at  the  end  of  each  period  to  the  principal,  the 
sum  of  the  principal  and  interest  becoming  the  new  principal. 

The  unit  of  time  is  1  year,  although  the  interest  may  be 
compounded  annually,  semiannually,  quarterly,  and  so  on. 

219.  If  15000  deposited  at  a  savings  bank  draws  interest 
at  4%,  semiannually,  the  interest  due  at  the  end  of  the  first 
half  year  will  be  2%  of  $5000,  or  $100. 

If  this  $100  is  not  drawn,  it  is  placed  to  the  credit  of  the 
depositor,  who  has  now  $5100  on  deposit. 

The  interest  for  the  second  half  year  is  2%  of  $5100,  or  $  102. 

If  this  is  not  drawn,  it  is  placed  to  the  credit  of  the  de- 
positor, making  his  deposit  $5202. 

The  interest  for  the  third  half  year  is  2%  of  $5202,  or 
$104.04. 

If  this  is  not  drawn,  it  is  placed  to  the  credit  of  the  de- 
positor, making  his  deposit  $5306.04  at  the  end  of  1  yr.  6  mo. 

Thus  $5000  at  4%  interest,  compounded  semiannually, 
will  in  1  yr.  6  mo.  amount  to  $5306.04;  and  the  compound 
interest  for  that  time  will  be  $5306.04  -  $5000  =  $306.04. 

$  5000  original  principal 

^ 

100.00  first  interest 
5000 


$  5100  amount  at  the  end  of  the  first  period 

^ 

102.00  second  interest 
5100 


$  5202  amount  at  the  end  of  the  second  period 

^2 

104.01  third  interest 
5202 


$  5306.04  amount  at  the  end  of  the  third  period 


282  ARITHMETIC 

220.   Find  the  compound  interest  on  1 5000  for  1  jr.  10  mo. 
15  da.  at  4%,  payable  semiannually. 

As  in  the  last  paragraph,  find  the  amount  of  $5000  for 

1  yr.  6  mo.,  and  then  complete  the  work  thus  : 

The  rate  per  cent  for  4  mo.  15  da.  or  f  yr.  =  |  x  4  %  =  li%. 

$  5306.04  amount  at  the  end  of  the  third  period 
.01| 
265302 
530604 


79.5906  fourth  interest 
5306.04 


$  5385.68  amount  at  the  end  of  the  fourth  period 
.-.  the  compound  interest  =  $  5385.63  -  $  5000  =  $  385.63. 
Or  thus,  find  the  interest  on  $5306.04  for  1  yr.  at  4%,  then  take  f  of  it. 

Exercise  173 
Find  the  amount  and  the  compound  interest  of : 

1.  $800  for  3  yr.  at  5%,  compounded  annually. 

2.  $425  for  4  yr.  at  4%,  compounded  annually. 

3.  $250  for  2  yr.  at  6%,  compounded  semiannually. 

4.  Find  the  amount  and  also  the  compound  interest  on  $  1000 
for  3  yr.  at  5%. 

5.  How  do  you  find  the  amount  of  a  siun  of  money  for  2  yr. 
at  6%  interest,  payable  semiannually? 

6.  Find  the  amount  of  $360  for  2  yr.  at  6%,  interest  payable 
semiannually. 

7.  Find  the  amount  of  $650  for  1  yr.  3  mo.,  interest  payable 
quarterly  at  4%  per  aimum. 

8.  Find  the  compound  interest  on  $8240  for  1  yr.  6  mo.  at 
5%,  payable  semiannually. 

9.  State  how  to  find  the  amount  of  a  sum  of  money  at  com- 
pound interest,  for  a  given  time  and  rate. 

10.    Find  the  amount  and  also  the  compound  interest  on  $2500 
for  1  yr.  10  mo.  15  da,  at  6%,  payable  semiannually. 


EXCHANGE  283 

11.  Find  the  difference  between  the  interest  on  f  1050  for 
1  yr.  at  4%,  and  1  yr.  at  4%  compounded  quarterly. 

12.  Find  the  amount  of  $2000  in  2  yr.  at  6%,  compounded 
annually. 

13.  A  man  deposits  in  the  savings  bank  $1500,  on  which  the 
interest  at  3%  per  annum  is  to  be  added  to  the  principal  every 
6  mo.  How  much  money  has  the  man  in  the  bank  at  the  end 
of  2  yr.  ? 

14.  What  will  be  the  amount,  compound  interest,  of  $2400  for 
1^  yr.  at  6%  per  annum,  paid  half-yearly  ? 

EXCHANGE* 

221.  If  A  of  Chicago  owes  B  of  St.  Paul  a  sum  of  money, 
he  can  discharge  the  debt  in  any  one  of  several  ways.  He 
can  buy  a  post-office  order  at  the  Chicago  post-office  payable 
to  B  at  the  post-office  in  St.  Paul;  he  can  buy  an  express 
order  at  the  office  of  an  express  company,  payable  to  B  at 
any  office  of  the  same  company ;  or  he  can  buy  a  draft  at  a 
bank  payable  to  B  at  a  bank  in  St.  Paul. 

Give  some  reasons  why  it  is  better  to  discharge  a  debt  by 
means  of  a  post-office  order,  express  order,  or  draft  than  by 
sending  the  money  in  a  registered  letter  or  by  express  or 
check. 

222.  The  following  are  the  rates  charged  for  express  orders 
to  any  part  of  the  United  States  or  Canada : 

Rates  for  orders  not  over 


12.50 

3^ 

$40.00 

15^ 

5.00 

5^ 

50.00 

18^ 

10.00 

8^ 

60.00 

18^ 

20.00 

10^ 

75.00 

23^ 

30.00 

15^ 

100.00 

28^ 

*  For  Stocks  and  Bonds,  see  Chapter  XXI. 


284  ARITHMETIC 

Over  f  100  at  above  rates. 

Single  express  orders  are  not  issued  for  more  than  f  50, 
and  for  larger  amounts  additional  orders  are  issued. 

223.  The  fees  charged  for  post-office  orders  to  any  part  of 
the  United  States,  Porto  Rico,  and  the  Philippines  are  the 
same  as  for  express  orders  up  to  ^50.  The  fee  on  an  order 
not  exceeding  $60  is  20^,  175  is  25^,  1100  is  30^.  Single 
post-office  orders  are  not  issued  for  more  than  ilOO,  and  for 
larger  amounts  additional  orders  are  issued. 

Exercise  174 

1.  What  is  the  cost  of  an  express  order  for  $25?  $44? 
$73?     $78? 

2.  What  is  the  cost  of  a  post-office  order  for  $80?  $32? 
$95?     $1.50? 

3.  What  is  the  cost  of  an  express  order  for  $  75  ?     $  100  ? 

4.  What  is  the  cost  of  a  post-office  order  for  $  75  ?     $  100  ? 

5.  What  is  the  cost  of  a  draft  for  $75?  $100?  $150? 
$240?  $325?  $180?  The  charge  in  each  case  is  \%  and 
the  least  charge  25^. 

6.  By  which  of  the  three  methods  given  in  questions  3,  4,  and  5 
is  it  cheaper  to  send  money  in  sums  greater  than  $  75  ?  In  sums 
less  than  $75,  if  25^  is  the  smallest  charge  for  a  draft? 

7.  What  is  the  cost  of  a  draft  for  $  87.50  ?  $  120  ?  $  175  ? 
$287.50?  $192.80?  The  charge  in  each  case  is  J%,  and  the 
least  charge  is  25  ^. 

224.  Exchange  is  generally  conducted  through  bankers, 
who  issue  drafts  directing  a  second  bank  to  pay  a  specified 
sum  of  money  to  the  order  of  the  person  named  in  the  draft. 

A  Time  Draft  is  one  payable  at  a  specified  time  after  sight 
or  date. 


EXCHANGE  285 

If  A  in  Chicago  owes  B  in  St.  Paul  a  sum  of  money,  B  may  send  a  draft  to 
A  for  the  amount.  If  A  accepts  the  draft,  he  writes  the  word  "accepted" 
with  the  date  across  the  face  and  signs  liis  name. 

Exchange  is  the  system  of  paying  debts  to  persons  in  dis- 
tant places  without  actually  sending  the  money,  by  means 
of  money  orders  and  drafts. 

225.  (1)  Find  the  cost  of  a  draft  on  New  York  for  1600, 
when  exchange  is  ^  %  premium. 

The  premium  =  \%  of  $600  =  $1.50. 
.-.  the  cost  =  $600 +  $1.50  =  $601. 50. 

(2)  Find  the  cost  of  a  draft  on  New  Orleans  for  $  1200, 
payable  60  da.  after  date,  exchange  being  \  %  discount,  and 
interest  6  % . 

The  discount  =  \  %  of  $  1200  =  $  3.00. 
The  discount  for  63  da.  =  6  %  of  $  1200  for  63  da.  =  $  12.60. 
.  •.  the  cost  =  $  1200  -  $  3.00  -  $  12.60  =  $  1184.40. 

Show  that  if  exchange  had  been  \  %  premium,  the  cost  would  have  been 
$  1190.40.      Why  63  days  ?     See  §  213. 

Exercise  175 

1.  Find  the  cost  of  a  draft  for  ^  900  at  J%  premium. 

2.  Find  the  cost  of  a  draft  for  $  1600  at  |%  discount. 

3.  Find  the  cost  of  a  draft  for  $  4500  at  |%  discount. 

4.  Find  the  cost  of  a  draft  for  %  2800  at  \oi\f^o  premium. 

5.  Find  the  cost  of  a  draft  for  $1000,  payable  in  60  da., 
exchange  being  ^%  premium,  and  interest  6%. 

6.  Find  the  cost  of  a  draft  for  $360,  payable  in  30  da., 
exchange  being  |%  discount,  and  interest  5%. 

7.  Find  the  cost  of  a  draft  for  $1250,  payable  in  60  da., 
exchange  being  ^%  premium,  and  interest  4|%. 


286  ARITHMETIC 

8.  Find  the  cost  of  a  draft  for  ^  1800,  payable  in  30  da.,  when 
exchange  is  at  par,  and  interest  4%. 

9.  Find  the  cost  of  a  bill  of  exchange  on  London  for  £  600, 
when  exchange  is  quoted  at  $  4.88. 

10.  Find  the  cost  of  a  60-da.  draft  on  Liverpool  for  £  750, 
exchange  at  60  da.  being  $  4.86. 

11.  What  is  the  cost  of  a  bill  of  exchange  in  Paris  for  2400 
francs  at  5.16|  francs  per  ^  1  ? 

12.  What  is  the  cost  of  a  bill  of  exchange  on  Berlin,  for  2400 
marks,  the  rate  of  exchange  being  95^  ^  for  4  marks  ? 

13.  August  21, 1899,  wheat  was  reported  6  d.  per  bushel  higher 
in  London  than  on  the  previous  day.  Find  in  dollars  and  cents 
the  increase  in  price  on  100  bu.  {£1  =  $  4.86|), 

14.  August  22, 1899,  wheat  was  reported  16  centimes  per  bushel 
higher  in  Paris  than  on  the  previous  day.  Find  in  dollars  and 
cents  the  increase  in  price  on  100  bu.  (1  franc  =  19.4^). 

15.  The  White  Star  Liner  Oceanic  was  open  to  the  public  at 
Belfast,  Ireland,  August  19,  1899,  the  charge  being  2  s.  6  d  This 
is  how  many  cents  ?  (Is.  =  25^). 


CHAPTER   XVI 

RATIO   AND   PROPORTION 

226.  If  two  quantities  are  expressed  in  terms  of  the  same 
unit,  their  Ratio  is  the  quotient  obtained  by  dividing  the  num- 
ber measuring  the  first  quantity  by  the  number  measuring 
the  second  quantity. 

Thus  the  ratio  of  $  3  to  ^  5  =  |,  or,  as  it  is  frequently 
written,  3  :  5. 

The  first  term  of  a  ratio  is  called  the  Antecedent,  and  the 
second  the  Consequent. 

Since  a  ratio  may  he  expressed  as  a  fraction^  both  terms 
of  a  ratio  may  he  multiplied  or  divided  hy  the  same  number 
without  changing  its  value. 

Thus  8  :  12  =  2  : 3,  dividing  each  term  by  4. 

3:4  =  15:  20,  multiplying  each  term  by  5. 
3f :  4^  =  9  :  10,  multiplying  each  term  by  12  and  dividing  by  5. 

227.  Reduce  the  following  to  equivalent  ratios  by  multi- 
plication or  division,  and  write  your  results  as  in  the  preced- 
ing paragraph  : 

6:8  15:25  24:18  14:21 

2:3  5:4  5:7  8:9 

228.  A  Proportion  *  is  the  equality  of  two  or  more  ratios. 
Thus  6  :  8  =  9  :  12  is  a  proportion.     Each  of  the  two  ratios 

is  equal  to  3  : 4  or  |. 

*  For  Compound  Proportion,  see  Chapter  XXI. 

287 


288  ARITHMETIC 

229.  The  first  and  fourth  terms  of  a  proportion  are  called 
the  Extremes,  and  the  second  and  third  terms  the  Means. 

230.  In  the  following  proportions  show  that  the  product 
of  the  means  is  equal  to  the  product  of  the  extremes. 

6:8  =  9: 12  7: 14  =  3:6 

10:15  =  8:12  f  :|  =  8:9 

6:9  =  4:6  2J  :  3f  =  4  :  6 

231.  (1)  Find  the  value  of  x  in  the  proportion  10 :15= 2; :  36. 

lSx  =  360 ; 
a;  =  24. 
Show  by  §  230  that  24  is  the  correct  answer. 

(2)  Find  the  value  of  x  in  the  proportion  | :  6  =  ^ :  2;. 

3a;_24 
4    "  5  ' 

15  X  =  96. 

Multiplying  each  side  of  the  equation  by  20, 

Exercise  176 
Pind  the  value  of  x  in  the  following  proportions : 


1. 

6:8  =  a;:12. 

6. 

|:a;  =  2:6. 

2. 

9 :  6  =  18  :  a;. 

7. 

8:a:  =  f:6. 

3. 

8:a;  =  12:16. 

8. 

x:9  =  f:f 

4. 

a;:14  =  24:16. 

9. 

f  :f  =  a;:10. 

5. 

a;:5  =  9:15. 

10. 

2.1  :2V  =  10: 

RATIO   AND   PROPORTION  289 

232.  If  the  antecedent  and  consequent  of  a  ratio  are 
interchanged,  the  resulting  ratio  is  called  the  reciprocal  of 
the  given  ratio. 

Thus,  the  reciprocal  of  the  ratio  3  :  4  is  the  ratio  4  :  3. 

Write  the  reciprocals  of  the  following  ratios :  2:3;  4:7; 
0^:^',  21:35;  4:^:;  a;:8. 

233.  (1)  If  15  bbl.  of  flour  cost  163,  what  will  35  bbl. 
cost  ? 

Let  X  =  the  number  of  dollars  in  the  cost  of  35  bbl. 
Then,  15  :  35  =  63  :  x  ; 

15  a;  =  35  X  63  ; 

15 
.-.  35  bbl.  will  cost  ^  147. 

(2)  If  ^Q  men  can  do  a  piece  of  work  in  21  da.,  how  long 
will  it  take  24  men  to  do  it  ? 

Let  X  =  the  number  of  days  24  men  will  take. 
Then,  56  :  24  =  x  :  21. 

24  X  =  21  X  56  ; 

24 

.*.  24  men  will  take  49  days. 

In  this  solution  why  use  the  reciprocal  ratio  x :  21  instead  of  the  ratio 
21  :x? 

In  the  following  exercise  be  careful  to  note  whether  you  should  use  the 
direct  or  the  reciprocal  ratio  as  one-half  of  the  proportion. 

Exercise  177 

1.  Solve  the  examples  in  Exercise  &Q,  from  4  through  20,  using 
X  to  represent  the  unknown  quantity. 

2.  If  6  articles  cost  $14.30,  how  much  will  13  cost  at  the 
same  rate  ? 

u 


290  ARITHMETIC 

3.  If  25  lb.  of  tea  cost  $  10,  how  many  pounds  can  be  bought 
for  f  50  ? 

4.  If  a  loaf  of  bread  costs  11  ^  when  flour  is  $  0  a  barrel,  find 
its  cost  when  flour  is  $  1\  a  barrel. 

5.  A  bankrupt  owes  f  3000;  his  assets  are  ^1740.  What 
sum  will  a  creditor  receive  whose  claim  is  $  350  ? 

6.  The  expense  of  carpeting  a  room  was  f  100 ;  if  the  breadth 
of  the  room  had  been  4  ft.  greater,  the  expense  would  have  been 
$  120.     Find  the  breadth. 

7.  If  a  man  working  9f  hr.  per  day  finishes  a  piece  of  work  in 
0  da.,  in  what  time  would  he  have  finished  it  if  he  had  worked 
8  J  hr.  per  day  ? 

8.  A  garrison  of  1500  men  has  provisions  for  13  mo. ;  how 
long  will  their  provisions  last  if  it  is  increased  to  2200  men  ? 

9.  If  4  men  or  0  women  can  do  a  piece  of  work  in  20  da.,  how 
long  will  it  take  3  men  and  15  women  to  do  the  same  work  ? 

10.  A  creditor  receives  %  1.50  for  every  ^  4  of  what  was  due 
to  him,  and  thereby  loses  $  301.05.     Wiiat  was  the  sum  due  ? 

11.  In  a  certain  business  one  partner,  whose  share  is  f\  of  the 
whole,  receives  from  it  a  profit  of  f  859.20.  What  share  is  owned 
by  another,  whose  profit  is  f  1909  ? 

12.  A  person  contracts  to  do  a  piece  of  work  in  30  da.,  and 
employs  15  men  upon  it ;  the  work  is  half  finished  in  24  da.  How 
many  additional  workmen  must  be  then  introduced  in  order  to 
perform  the  contract  ? 

13.  The  profits  of  a  garden  for  2  yr.  were  $1450;  the  profits 
of  the  second  year  being  f|  of  those  of  the  first.  'Find  the  profits 
of  each  year. 

14.  If  10  men  can  do  a  piece  of  work  in  12  da.,  how  soon  after 
beginning  must  they  be  joined  by  3  more  so  as  to  finish  the  work 
in  10  da.  ? 


PROPORTIONAL  PARTS  291 

15.  Ii$  120  gain  $  5.81  in  126  da.,  find  the  gain  in  360  da. 

16.  A  bankrupt  who  is  paying  37 J^^  on  the  dollar  divides 
among  his  creditors  $  6300.     What  do  his  debts  amount  to  ? 

17.  If  3  men  or  5  boys  can  do  a  piece  of  work  in  18  da.,  in  how 
many  days  will  3  men  and  5  boys  do  a  piece  of  work  3  times  as 
great  ? 

18.  If  3  men  can  do  as  much  work  in  a  day  as  4  boys,  how 
long  will  it  take  64  boys  to  finish  a  piece  of  work  of  which  12  men 
have  done  ^  in  16  da.  ? 

19.  If  a  debt  after  a  deduction  of  3  %  becomes  f  1008.80,  what 
would  it  have  become  after  a  deduction  of  4  %  had  been  made  ? 

20.  Six  sheets  of  paper  measuring  8  in.  by  10  in.  weigh  an 
ounce.  Find  the  weight  of  120  sheets  of  the  same  kind  of  paper, 
each  sheet  measuring  6  in.  by  9  in. 

21.  A  person  walks  from  his  house  to  his  office  at  the  rate  of 
4  mi.  per  hour ;  but  finding  he  has  forgotten  something,  returns 
at  the  rate  of  5  mi.  per  hour.  Compare  the  time  spent  in  going 
with  that  spent  in  returning. 

22.  One  train  travels  8|-  mi.  in  20  min.,  and  a  second  train  9  mi. 
in  15  min.     Compare  their  rates  per  hour. 

23.  A  man  can  row  6  mi.  an  hour  in  still  water.  Compare  his 
rate  of  rowing  down  a  stream  which  flows  at  the  rate  of  2^  mi.  an 
hour  with  his  rate  of  rowing  up. 

PROPORTIONAL   PARTS 
234.    (1)  Divide  1720  in  parts  proportional  to  4,  5,  and  6. 

The  total  number  of  parts  =  4  +  5  +  6  =  15. 

.-.  the  first  part  =  j\  of  $  720  =  $  192, 
the  second  part  =  j%  of  $  720  =  $  240, 
the  third  part     =  ^%  of  .|  720  =  $  288. 


292  ARITHMETIC 

Or  thus : 

Let  X  =  the  number  of  dollars  in  the  value  of  one  share. 
Then  ix -\-  bx -\- 6x  =  720; 

15  X  =  720 ; 
x  =  48. 
4  a;  =  192,  5  a;  =  240,  6x  =  288. 

.-.  the  parts  are  $  192,  $  240,  $  288. 

(2)  Divide  316  lb.  into  parts  proportional  to  ^,  ^,  and  ^. 

Multiplying  ^,  ^,  and  J  by  their  L.  C.  M.  120,  we  have  the  parts  propor- 
tional to  40,  24,  and  15. 

The  total  number  of  parts  =  40  +  24  +  15  =  79. 

.-.  the  parts  are  respectively  f §,  ff,  and  j|  of  316  lb.  =  160,  96,  and  60  lb. 

Proof.  —  Dividing  160,  96,  and  60  by  480,  the  denominator  which  reduces 
160  to  ^,  we  have  ^,  |,  |,  which  proves  the  results  found  to  be  correct. 

Or  thus : 

Let  X  =  the  number  of  pounds  in  one  share. 


Then 

XXX 

3     6      8 

=  316; 

01 

'  multiplying  by 

120, 

40  X  +  24  a;  +  15  x  : 

=  37920 ; 

79x: 

=  37920 ; 

X 

=  480. 

X 

3 

160, 

1  =  96. 

X 

8 

.-. 

the  parts  are  160  lb.. 

96  lb.,  60  lb. 

=  60. 


Exercise  178 

1.  Divide  1331  into  parts  proportional  to  2,  4,  6. 

2.  Divide  $  73.50  into  parts  proportional  to  J,  f,  J. 

3.  Divide  19  T.  1104  lb.  into  parts  proportional  to  ^,  J,  J. 

4.  Divide  $  1064  into  parts  proportional  to  2,  2^,  2|. 


PROPORTIOXAL  PARTS  293 

6.    Divide  180  lb.  into  parts  proportional  to  3.3,  .7,  .5. 

6.  Divide  $  4500  between  two  persons  in  proportion  to  their 
ages,  which  are  21  and  24  yr. 

7.  Two  men  receive  $  15  for  doing  a  certain  piece  of  work. 
Now  one  man  had  worked  only  3  da.,  while  the  other  had  worked 
5  da.  on  the  job.  If  the  money  is  to  be  divided  in  proportion 
to  the  lengths  of  time  the  men  worked,  how  much  should  each 
receive  ? 

8.  Divide  4472  into  parts  which  shall  be  to  each  other  in  the 
ratio  of  3,  5,  7,  11. 

9.  Divide  $  84.42  into  two  parts  which  shall  be  to  each  other 
as  5  :  16. 

10.  A  company  of  militia  consisting  of  72  men  is  to  be  raised 
from  three  towns  which  contain  respectively  1500,  7000,  and  9500 
men.     How  much  must  each  town  provide? 

11.  Sugar  is  composed  of  49.856  parts  oxygen,  43.625  carbon, 
and  6.879  hydrogen.  How  many  pounds  of  each  are  there  in 
1300  lb.  of  sugar? 

12.  Gunpowder  is  composed  of  nitre,  charcoal,  and  sulphur  in 
the  proportion  of  33,  7,  and  5. 

(1)  How  many  pounds  of  sulphur  are  there  in  180  lb.  of  powder  ? 

(2)  How  many  pounds  of  powder  can  be  made  with  30  lb.  of 
sulphur  ? 

(3)  How  much  nitre  and  sulphur  must  be  mixed  with  112  lb. 
of  charcoal  to  form  gunpowder  ? 

13.  A  man  divides  $  3300  amongst  his  three  sons,  whose  ages 
are  16,  19,  and  25  yr.,  in  sums  proportional  to  their  ages ;  2  yr. 
afterwards  he  similarly  divides  an  equal  sum,  and  again  after 
3  yr.  more.     How  much  does  each  receive  in  all  ? 

14.  Two  persons  travelling  together  agree  to  pay  expenses  in 
the  ratio  of  7  to  5.  The  first  (who  contributes  the  greater  sum) 
pays  on  the  whole  $  103.40,  the  second  $  63.40.  What  must  one 
pay  the  other  to  settle  their  expenses  according  to  agreement  ? 


294  ARITHMETIC 

15.  Divide  $  480  among  A,  B,  C,  and  D,  so  that  B  may  receive 
as  much  as  A ;  C  as  much  as  A  and  B  together ;  and  D  as  much 
as  A,  B,  and  C  together. 

PARTNERSHIP* 

235.  In  Simple  Partnership  the  capital  of  each  partner  is 
supposed  to  be  invested  for  the  same  time. 

236.  A,  B,  and  C  engage  in  business.  A  furnishes  i  7500, 
B  15000,  and  C  14500.  If  they  gain  12380,  what  is  each 
one's  share  ? 

Dividing  their  capitals  by  $  500,  we  have  their  capitals,  and  therefore  their 
gains  proportional  to  15,  10,  and  9. 

The  total  number  of  parts  =  15  +  10  +  9  =  34. 

.-.  their  respective  gains  are  ^|,  |f,  and  y\  of  $2380  =  31050,  $700,  and 
$630. 

Or  thus : 

Let  X  =  the  number  of  dollars  gain  on  $  100  capital. 

Then  75  a;  +  50  x  +  45  x  =  2380  ; 

170  X  =  2380  ; 

X  =  14. 

75  a;  =  1050,  50  a;  =  700,  45  a;  =  630. 

.-.  their  respective  gains  are  $  1050,  $  700,  $  630. 

Exercise  179 

1.  Two  merchants,  A  and  B,  form  a  joint  capital.  A  puts  in 
$  1200  and  B  $  1800.  They  gain  $400.  How  ought  the  gain  to 
be  divided  between  them  ? 

2.  A  bankrupt  owes  three  creditors,  A,  B,  and  C,  $  175,  $  210, 

and  $265,  respectively;  his  property  is  worth  $422.50.     What 
ought  each  to  receive  ? 

«  For  Compound  Partnership,  see  Chapter  XXI. 


PARTNERSHIP  295 

3.  A,  B,  and  C  entered  into  partnership.  A  puts  in  $  6000, 
B  $4000,  and  C  f  2000.  They  gained  $2250.  What  is  each 
one's  share  of  the  gain  ? 

4.  Two  men  purchase  a  house  for  f  3600,  the  first  contribut- 
ing $1600  and  the  second  $  2000.  If  it  rents  so  as  to  pay  12% 
on  its  value,  what  share  of  the  rent  should  each  receive  ? 

5.  Two  persons  have  gained  in  trade  $3456;  one  put  in 
$10,560  and  the  other  $8640.  What  is  each  person's  share  of 
the  profits  ? 

6.  R.  Stuart  and  G.. Armstrong  enter  into  partnership.  Stuart 
contributes  $  4500  to  the  partnership  and  Armstrong  contributes 
$7500.  Their  net  gain  at  the  end  of  the  year  is  $1750.  How 
much  of  this  sum  should  each  partner  receive  ? 

7.  Three  partners  invest  respectively  $7800,  $5750,  and 
$  9450  in  business.  At  the  end  of  the  first  year  they  find  their 
net  gain  to  be  $3156.  What  is  the  amount  of  each  partner's 
share  of  this  gain  ? 

8.  A,  B,  and  C  form  a  partnership  with  a  capital  of  $  20,000. 
A  contributes  $  5000,  B  $  7000,  and  C  the  remainder.  They  gain 
20%  of  the  total  capital.     Find  each  man's  share  of  the  profits. 

9.  T.  Allan  and  E.  Jamieson  engage  in  business  with  a  joint 
capital  of  $  19,200,  and  agree  to  share  gains  and  losses  in  propor- 
tion to  their  investments.  At  the  end  of  a  year  Allan  receives  a 
dividend  of  $  1100,  and  Jamieson  a  dividend  of  $  1300.  What 
was  the  amount  of  the  investment  of  each  ? 

10.  D.  Rowan,  F.  Galbraith,  and  J.  Munro  enter  into  partner- 
ship. They  gain  $  7500,  of  which  Rowan  receives  $  2100,  Gal- 
braith $3100,  and  Munro  the  balance.  How  much  did  Rowan 
and  Galbraith  respectively  invest  if  the  amount  of  Munro's  invest- 
ment was  $18,000? 

11.  A,  B,  and  C  pay  $37.80  as  rent  for  a  pasture.  A  puts  in 
5  horses,  B  12  cows,  and  C  60  sheep.  If  1  horse  eats  as  much  as 
2  cows,  and  1  cow  as  much  as  3  sheep,  what  rent  should  each  pay  ? 


CHAPTER  XVII 


SQUARE  ROOT* 

237.    (1)  Find  the  square  root  of  17.3056. 


17.30'56 
16 

|4.16 

81 

130 
81 

826 

4956 
4956 

To  prove  4.16  the  right  answer,  square  4.16,  and  the  result  will  be  found 
to  be  17.3056. 

(2)  Extract  the  square  root  of  35  to  three  decimal  places. 


35  1  5.916 

25 

109 

1000 

981 

1181 

1900 

1181 

11826 

71900 

70956 

(3)  Extract  the  square  roots  of  ||,  ||,  |. 

l25^\/26^5 
>49      V49     7' 

/35^V35^  5.916^  845 
>'49     V49         7         ■       • 

\/i=  V:625  =  .7905. 

Which  denominator  is  not  a  perfect  square  ?    Why  reduce  |  to  a  decimal 
before  extracting  the  square  root  ? 

*  Review  Chapter  IX.     For  Cube  Root,  see  Chapter  XXI. 
296 


SQUARE   ROOT  297 

Exercise  180 
Find  the  square  root  of : 

1.  40.96;  65.61;  2.1025. 

2.  167.9616 ;  28.8369  ;  57648.01. 

3.  .042849;  .00139876;  .00203401. 

4.  5774409;  5.774409. 

5.  10.3041;  2321.3124;  .0050367409. 

6.  2 ;  20  ;  .4 ;  1000  to  four  decimal  places. 

7       144  .     S2A-    fil 

ft      901  .156.1.     2.109. 
».     ^U^  ,    l^iQQ,     3  )     98T71* 

Q       3..     5.-    J_ 

^-      8  >     9  '     1  1" 


•     CHAPTER   XVIII 

MENSURATION 

238.   The  rectangle  has  been  treated  of  in  preceding  para- 
graphs. 

Exercise  181 

1.  What  is  the  area  of  a  rectangle  15  ft.  long  and  12  ft.  wide  ? 

2.  What  is  the  area  of  a  rectangular  field  15  chains  long  and 
6  chains  wide  ?     How  many  acres  does  it  contain  ? 

3.  How  many  acres  in  a  field  16  chains  long  and  10  chains 
wide? 

4.  How  many  acres  in  a  field  32  rd.  long  and  25  rd.  wide  ? 

5.  A  rectangular  flower  bed  6  ft.  long  and  4  ft.  wide  is  sur- 
rounded by  a  walk  1  ft.  6  in.  wide.  How  many  square  feet  in  the 
walk  ?  (Make  a  drawing  to  represent  the  bed  and  walk,  on  the 
scale  of  1  in.  to  1  ft.) 

6.  A  garden  50  ft.  long  and  40  ft.  wide  is  surrounded  by  a  walk 
3  ft.  wide.     How  many  square  feet  in  the  walk  ? 

7.  Make  problems  similar  to  examples  5  and  6. 

8.  Find  the  value  of  a  field  60  rd.  long  and  40  rd.  wide,  at  $  75 
an  acre. 

9.  A  town  lot  4  rd.  wide  and  6  rd.  deep  sold  for  $  1200.  What 
is  that  per  acre  ? 

10.  A  rectangular  room  is  18  ft.  long  and  12  ft.  wide.     How 
much  smaller  is  it  than  a  square  room  of  equal  perimeter  ? 

11.  A  rectangular  room  is  16  ft.  long  and  9  ft.  wide.     Find  the 
length  of  the  side  of  a  square  room  of  equal  area. 

298 


MENSURATION  299 

12.  A  square  field  contains  21  A.  Find  the  length  of  a  side  of 
the  field  in  chains. 

13.  Find  the  cost  of  painting  a  surface  19  ft.  6  in.  by  83  ft. 
4  in.  at  5p  a  square  foot. 

14.  A  square  field  contains  exactly  8  A.  Determine  the  length 
of  a  side  of  the  field  in  chains  and  links. 

15.  The  area  of  a  chess-board  marked  in  8  rows  of  8  squares 
each  is  100  sq.  in.     Find  the  length  of  a  side  of  a  square. 

16.  On  a  certain  map  an  area  of  16  sq.  mi.  is  represented  by 
9  sq.  in.     What  part  of  an  inch  represents  a  mile  ? 

17.  On  a  certain  map  it  is  found  that  an  area  of  25  sq.  mi.  is 
represented  by  an  area  of  6.25  sq.  in.  Give  the  scale  of  the  map 
in  miles  to  the  inch. 

18.  A  rectangle  measures  18'  by  30'.  Find  the  difference  be- 
tween its  area  and  that  of  a  square  of  equal  perimeter. 

19.  Two  rectangular  fields  are  of  equal  area.  One  field  meas- 
ures 15  ch.  by  20  ch.;  the  other  is  square.  Find  the  length  of  a 
side  of  the  latter  field,  correct  to  the  nearest  link. 

20.  How  many  stalks  of  wheat  could  grow  on  1  sq.  yd.  of 
ground,  allowing  each  stalk  a  rectangular  space  of  2"  by  3"? 
How  many  on  1  A.  ? 

21.  How  many  pieces  of  turf  3'  6"  by  1'  3"  will  be  required  to 
sod  a  rectangular  lawn  28'  by  60'  ? 

22.  Sidewalks  4  ft.  wide  are  laid  on  both  sides  of  a  street  440 
yd.  long.  Find  the  cost  of  the  sidewalks  at  $1.35  per  square 
yard  for  the  pavement  and  75  ^  per  lineal  yard  for  curbing. 

23.  A  board  containing  6  sq.  ft.  is  9  in.  wide.    Find  its  length. 

24.  What  length  must  be  cut  off  a  board,  which  is  7^  in.  broad, 
so  that  the  area  may  contain  3  sq.  ft.  ? 


300 


ARITHMETIC 


239.   A  Quadrilateral  is  a  plane  figure  having  four  sides. 
A  Parallelogram  is  a  quadrilateral  whose  opposite  sides 
are  parallel. 


D  C 

240.  To  find  the  area  of  a  parallelogram  : 

Let  perpendiculars  be  drawn  from  C  and  D  perpendicular  to  AB.  Then 
it  is  evident  that  the  triangles  marked  a  are  equal.  Adding  to  each  the 
quadrilateral  marked  c,  it  is  evident  that  the  parallelogram  ABCD  is  equal 
to  the  rectangle  upon  the  base  CD. 

Henee^  to  find  the  measure  of  the  area  of  a  parallelogram^ 
multiply  the  measure  of  its  base  by  the  measure  of  its  altitude, 

a  =  bh. 

241.  A  Trapezoid  is  a  quadrilateral  two  of  whose  sides 
are  parallel. 

The  parallel  sides  are  called  Bases  and  the  perpendicular 
distance  between  the  two  bases  is  called  the  Altitude. 


D         E  C 

Thus,  in  the  trapezoid,  AB  and  CD  are  the  bases,  and  AE 
the  altitude. 


MENSURATION 
242.   To  find  the  area  of  a  trapezoid ; 


301 


L      A 

B 

M 

/ 

\ 

K 

) 

/ 

\ 

\ 

I 

> 

D 

D 

1 

M 

Let  ABCD  be  a  trapezoid,  and  let  perpendiculars  be  drawn  through  E 
and  F,  the  middle  points  of  AD  and  BG,  to  AB  and  CD.  Then  it  is  evident 
that  the  triangles  a  and  a'  are  equal,  and  also  c  and  d. 

To  a'  and  c'  and  also  to  a  and  c  add  the  figure  ABFNOE,  and  we  have 
the  trapezoid  equal  to  the  rectangle  LMNO. 


Again, 


Adding, 


EF=AB  +  AL-\-BM, 
EF=  CD  ~  DO  -  NC. 
2  EF:=zAB-^  CD, 
since  AL-  DO,  and  BM  =  NG, 


.-.  ON=\{AB+  CD). 
Therefore,  to  measure  the  area  of  the  trapezoid,  we  multiply  the  measure 
of  ON,  i.e.  oi^(AB  +  CD),  by  that  of  the  altitude. 

Hence  the  area  of  a  trapezoid  is  found  hy  multiplying  the 
measure  of  one-half  the  sum  of  its  parallel  sides  hy  the  measure 
of  its  altitude.  ,  ,^      ^,^^ 

243.   Find  the  area  of  a  trapezoid  whose  parallel  sides  are 
respectively  8  in.  and  4  in.  long  and  altitude  6  in. 
The  sum  of  the  bases  =  8  in,  +  4  in.  =  12  in. 
Half  the  sum  of  the  bases  =  6  in. 
The  altitude  =  4  in. 
.-.  the  area  =  6  x  4  sq.  in.  =  24  sq.  in. 

Draw  this  trapezoid  and  the  equivalent  rectangle.  Show  by  measurement 
that  the  base  of  the  rectangle  is  one-half  of  (8  +  4)  in. ,  or  6  in.  long. 


302  ARITHMETIC 

Exercise  182 

1.  Eind  the  area  of  a  parallelogram  whose  base  is  6  in.  and 
altitude  3  in.  Draw  this  parallelogram  and  the  equivalent 
rectangle. 

2.  Find  the  areas  of  the  following  parallelograms : 

Base  Altitude  Base  Altitude 

16  in.  9  in.  15  ch.  8  ch. 

12  ft.  10  ft.  20  rd.  16  rd. 

18  yd.  15  yd.  14  ch.  15  ch. 

3.  Find  the  number  of  acres  in  each  of  these  parallelograms : 

Base  Altitude  Base  Altitude 

90  rd.  80  rd.  60  rd.  18  rd. 

36  rd.  24  rd.  15  ch.  9  ch. 

25  ch.  6  ch.  63  ch.  21  ch. 

4.  Find  the  area  of  a  trapezoid  whose  bases  are  24  in.  and  20 
in.  and  altitude  12  in. 

Draw  this  trapezoid  and  the  equivalent  rectangle  on  the  scale 
of  4  in.  to  1  in. 


5.    Find  the  area 

of  each  of  these  trapezoids : 

Bases 

Altitude 

Bases 

Altitude 

20  in.       16  in. 

12  in. 

8  rd.       12  rd. 

6rd. 

15  ft.       10  ft. 

6  ft. 

6  rd.         4  rd. 

16^  yd. 

27  ft.       14  ft. 

9  ft. 

16  yd.       12  yd. 

4rd. 

6.    Find  the  number  of  acres 

in  each  of  these  trapezoids : 

Bases 

Altitude 

Bases 

Altitude 

100  rd.     60  rd. 

25  rd. 

17  ch.         8  ch. 

6  ch. 

75  rd.     45  rd. 

36  rd. 

142  yd.    100  yd. 

80  yd. 

25  ch.     15  ch. 

9ch. 

16  ch.      13  ch. 

14  ch. 

7.   Find  the  side  of  a  square  equal  in  area  to  a  trapezoid,  the 
bases  of  which  are  120  ft.  and  60  ft.  and  the  altitude  40  ft. 


MENSURATION 


303 


8.  The  length  of  the  base  of  a  parallelogram  is  45  ft. ;  the 
length  of  the  perpendicular  on  the  base  from  the  opposite  side 
is  28  ft.     Find  the  area. 

9.  The  lengths  of  the  parallel  sides  of  a  trapezoid  are  12  ft. 
and  17  ft.,  and  the  perpendicular  distance  between  these  sides  is 
8  ft.     Find  its  area. 

10.  The  parallel  sides  of  a  garden  are  84  yd.  and  92  yd.  respec- 
tively, and  their  perpendicular  distance  27|-  yd. ;  what  did  it  cost 
at  §1200  an  acre? 


244.  Let  ABC  be  a  triangle,  and  let  the  rectangle  BODE  be  drawn. 
Then  it  is  evident  that  the  triangles  a  and  a'  and  c  and  d  are  equal.  Hence 
the  triangle  ABC  is  one-half  of  the  rectangle  BCDE.  Hence,  to  find  the 
area  of  a  triangle,  multiply  one-half  the  measure  of  the  base  by  that  of  the 
altitude. 

a  =  \  bh. 


245.  To  find  the  area  of  a  triangle  when  the  lengths  of 
the  sides  are  given : 

Find  one-half  of  the  sum  of  the  measures  of  the  sides  ;  sub- 
tract from  this  the  measure  of  each  side  separately.  The 
square  root  of  the  product  of  these  four  results  will  give  the 
measure  of  the  area  of  the  triangle* 


304  ARITHMETIC 

Find  the  area  of  a  triangle  whose  sides  are  9  in.,  7  in.,  and 

6  in. 

28  =  9  +  7  +  6  =  22. 

s=ll,    8-9  =  2,    8-7=4,   8-6  =  5. 

Vll  .2.4.5  =  V440  =  20.9. 

.*.  the  area  =  20.9  sq.  in. 

Draw  this  triangle  and  see  if  its  area  seems  to  be  nearly  equal  to  that  of  a 

rectangle  7  in.  by  3  in.  (i.e.  21  sq.  in.). 


Exercise  183 

Find  the  area  of  each  of  these 

triangles : 

Base                         Altitude 

Base 

Altitude 

.2  in.                  8  in. 

80  rd. 

30  rd. 

9  ft.                  6  ft. 

15  ch. 

8ch. 

.6  yd.               12  yd. 

13  ch. 

5ch. 

2.  Find  the  area  of  a  triangular  piece  of  park  -whose  base  is 
8  rd.  and  altitude  5  rd.     What  is  its  value  at  $  12,000  an  acre  ? 

3.  Find  the  value  of  a  triangular  field,  whose  base  is  75  rd. 
and  altitude  48  rd.,  at  $  72  an  acre. 

4.  Find  the  altitude  of  a  triangle  whose  base  is  12  in.  and 
area  48  sq.  in.     (6x=  48.) 

5.  Find  the  altitude  of  each  of  these  triangles: 

Area  Base  Area  Base 

108  sq.  in.  18  in.  4  A.  16  ch. 

196  sq.  in.  28  in.  2^  A.  10  ch. 

2  A.  32  rd.  1  A.  110  yd. 

6.  Find  the  base  of  each  of  these  triangles : 

Area  Altitude  Area  Altitude 

36  sq.  in.  6  in.  4^  A.  10  ch. 

84  sq.ft.  14  in.  7^  A.  40  rd. 

lA.  16  rd.  l^A.  120  yd. 


MENSURATION 


305 


7.  Find  the  area  of  a  triangle  whose  sides  are  10  in.,  8  in., 
and  6  in.     Draw  this  triangle  and  test  your  result. 

8.  Find  the  areas  of  the  triangles  the  lengths  of  whose  sides 
are  respectively : 

(1)  13  yd.,  10  yd.,  and  13  yd.  (4)    8  in.,    7  in.,  and  5  in. 

(2)  13  yd.,  24  yd.,  and  13  yd.  (5)   9  in.,  12  in.,  and  7  in. 

(3)  13  ft.,     4  ft.,   and  15  ft.  (6)   3  in.,    5  in.,  and  7  in. 

9.  The  sides  of  a  triangular  piece  of  park  are  17  rd.,  15  rd., 
and  8  rd.     Find  its  value  at  $  2400  per  acre. 

10.  The  sides  of  a  triangular  field  are  41  rd.,  40  rd.,  and  9  rd. 
Find  its  value  at  $  96  per  acre. 

11.  Find  the  number  of  acres  in  a  triangular  field  whose  sides 
are  12  ch.,  16  ch.,  and  8  ch. 

246.   To  find  the  area  of  a  circle : 


Draw  a  circle  on  cardboard  and  cut  it  out  —  the  larger  the  better.  Divide 
each  half  of  the  circle  as  the  semicircle  in  the  figure  is  divided,  the  arcs  A,  B, 
C,  D,  etc. ,  being  as  nearly  equal  as  possible.  Cut  the  circle  into  two  equal 
parts  along  the  line  AOM. 


306  ARITHMETIC 

Cut  along  OB,  OC,  etc.,  cutting  nearly  to  the  points  By  C,  D,  but  not 
separating  tlie  parts  entirely  at  these  points.  Spread  the  resulting  figure  out 
as  in  the  darker  part  of  the  figure  below. 


A      B -C      D      £ 


Then  cut  up  the  other  semicircle  in  the  same  way  ;  spread  open  the  parts 
and  fit  the  two  semicircles  together,  as  in  the  figure.  The  resulting  figures 
will  be  nearly  a  rectangle.  The  smaller  the  arcs  AB,  BC,  etc.,  the  more 
nearly  the  area  will  be  to  a  rectangle  whose  base  is  equal  to  one-half  the 
circumference  and  whose  altitude  is  equal  to  the  radius  of  the  circle. 

Hence  tJie  measure  of  the  area  of  a  circle  is  one-half  the 
product  of  the  measures  of  the  circumference  arid  the  radius* 
It  may  also  he  expressed  thus : 

The  measure  of  the  area  =  |  cr. 

Again,  since  c=  3.1416  x  2r, 

the  measure  of  the  area  =  3.1416  r^. 

Both  formulas  are  useful. 

The  last  rule  may  be  read :  The  measure  of  the  area  of  a 
circle  is  found  hy  multiplying  the  square  of  the  measure  of  the 
radius  hy  3.1416. 

On  referring  to  Exercise  52,  example  16,  it  will  be  seen 
that  the  circumference  of  a  circle  is  equal  to  the  diameter  mul- 
tiplied hy  3.1416. 

Note.  —  y  may  be  substituted  in  the  above  formula  for  3.1416  when 
desired.    Why  ? 


MENSURATION^  307 

247.  (1)  Find  the  circumference  of  a  circle  whose  diame- 
ter is  6  in. 

The  circumference  =  3.1416  x  6  in.  =  18.8496  in. 

(2)  Find  the  diameter  of  a  circle  whose  circumference  is 
25  in. 

Let  cc  =  the  number  of  inches  in  the  diameter. 
Then  25  =  3.1416  x,  or  3.1416  a;  =  26. 

x  =  25 -3.1416  =  7.9. 

.*.  the  diameter  =  7.9  in. 

(3)  Find  the  area  of  a  circle  whose  diameter  is  12  in. 

The  area  =  3.1416  x  6^  sq.  in. 
=  3.1416  X  36  sq.  in. 
=  113.09  sq.  in. 

Exercise  184 

1.  Find  the  circumference  of  each  of  the  following  circles : 

Diameter  =  4  in.,  12  in.,  25  in.,    7  in.,  15  in. 
Badius      =  5  in.,    7  in.,  2^  in.,  4J  in.,    9  in. 

2.  Find  the  diameter  of  a  circle  whose  circumference  is  36  in. 

3.  Find  the  diameter  of  each  of  the  following  circles : 

Circumference  =  16  in.,  45  in.,  24  in.,  29  in. 

4.  Find  the  circumference  of  a  pond  whose  radius  is  12  ft.  6  in. 

5.  Lady  smith,  South  Africa,  when  besieged,  was  defended  by 
cannon  that  commanded  a  radius  of  4  mi.  Show  that  the  invest- 
ing line  must  have  been  at  least  25  mi.  in  circumference. 

6.  Find  the  area  of  each  of  the  following  circles : 

Radius  =  2  in.,  5  in.,  10  in.,  7  in.,  2i  in. 

7.  A  cow  is  tied  to  a  stake  with  a  chain  24  ft.  in  length. 
How  many  square  yards  can  she  graze  over  ? 


308 


ARITHMETIC 


8.  A  circular  pond  is  12  yd.  in  diameter.     Find  its  area. 

9.  A  circular  lake  32  rd.  in  diameter  has  a  road  around  it 
which  is  4  yd.  wide.  Find  the  number  of  square  yards  in  the 
road. 

10.  Find  the  total  pressure  on  a  plate  14  in.  in  diameter,  the 
pressure  per  square  inch  being  25  lb.     (Use  ■^-.) 

11.  Out  of  a  circle  of  radius  3  ft.  is  cut  a  circle  of  radius  2  ft. 
Find  the  area  of  the  remainder. 

12.  There  is  a  circular  fish-pond  of  90  ft.  radius,  surrounded 
by  a  walk  5  ft.  wide.     Find  the  area  of  the  walk. 

13.  Show  that  the  area  of  a  circle  whose  radius  is  35  in.  is 
equal  to  the  sum  of  the  areas  of  four  circles  of  10  in.,  15  in.,  18  in., 
and  24  in.  radius  respectively. 

14.  The  diameter  of  a  semicircle  is  10  in.     Find  its  perimeter. 


^71 


-A. 


248.   A  Prism  is  a  solid  whose  bases  are  equal  polygons 
and  lateral  faces  parallelograms. 

Fold  a  sheet  of  paper  into  a  prism  whose  base  is  (a)  a  triangle ;  (6)  a 
square  ;  (c)  a  pentagon.    Unfold  the  prisms  into  rectangles. 


249.  If  the  lateral  surface  of  a  right  prism  is  placed  in 
one  plane,  it  will  form  a  rectangle  whose  length  is  the  perim- 
eter (p)  of  prism  and  whose  width  is  the  altitude  (A)  of 
the  prism. 

To  find  the. area  of  the  lateral  surface  of  a  prism  multiply 
the  measure  of  the  perimeter  of  its  base  by  that  of  its  altitude, 

a  =  ph. 


MENSURATION  309 

To  find  the  volume  of  a  prism  multiply  the  measure  of  the 
area  of  its  base  by  that  of  its  altitude. 

V  —  hh. 

Exercise  185 

1.  Find  the  lateral  area  of  a  right  prism  the  perimeter  of  whose 
base  is  9  in.  and  altitude  6  in.  Make  this  prism  by  folding  a 
sheet  of  paper  9  in.  by  6  in.  parallel  to  the  6  in.  side. 

2.  Find  the  lateral  area  of  each  of  these  right  prisms : 

Perimeter  of  base  =  24  in.,  36  in.,  2  ft.  10  in. 
Altitude  =    8  in.,  15  in.,  1  ft.  6  in. 

3.  Find  the  lateral  area  of  a  square  prism  whose  altitude  is 
16  ft.  and  the  side  of  whose  base  is  9  ft.     Find  its  total  area. 

4.  Find  the  lateral  area  of  a  triangular  prism  whose  altitude  is 
20  in.  and  the  sides  of  whose  base  are  13  in.,  17  in.,  and  11  in. 

5.  Find  the  total  area  of  a  right  triangular  prism  whose  alti- 
tude is  75  in.  and  the  sides  of  whose  base  are  51  sq.  in.,  45  in.,  and 
24  in.  long. 

6.  Find  the  total  area  of  a  right  triangular  prism  whose  alti- 
tude is  6  in.  and  whose  base  is  an  equilateral  triangle  of  side  4  in. 

7.  Find  the  volume  of  a  prism  whose  base  contains  25  sq.  in. 
and  whose  altitude  is  7  in. 

8.  Find  the  volume  of  each  of  these  prisms : 

Area  of  base  =  9  sq.  in.,  28  sq.  in.,  3  sq.  ft. 
Altitude  =  4  in.,  10  in.,  1  ft.  9  in. 

9.  Find  the  volume  of  each  of  these  prisms  : 

(1)  Base  a  square  of  side  5  in. ;  altitude  1\  in. 

(2)  Base  a  triangle  of  base  6  in.  and  altitude  3  in. ;  altitude  4  in. 

(3)  Base  a  triangle,  sides  51  in.,  45  in.,  24  in. ;  altitude  2  ft.  2  in. 

(4)  Base  an  equilateral  triangle,  side  10  in. ;  altitude  7  in. 

(5)  Base  a  rectangle,  sides  9  in.,  6  in. ;  altitude  5|-  in. 

(6)  Base  a  parallelogram,  length  11  in.,  width  6  in. ;  altitude  8  in. 


310  ARITHMETIC 

250.  If  we  wrap  a  rectangular  sheet  of  paper  about  a  right 
circular  cylinder,  we  find  that  the  area  of  the  curved  surface 
of  the  cylinder  is  a  rectangle  whose  base  is  the  circumference 
of  the  cylinder  and   altitude  the   height  of   the   cylinder. 

Hence  the  measure  of  the  area  of  the  curved  surface  of  a 
right  circular  cylinder  is  equal  to  the  product  of  the  measure 
of  the  perimeter  of  the  base  and  that  of  the  altitude. 

a  =  ph. 
What  must  we  add  to  this  to  find  the  total  area  of  the 
cylinder  ? 

To  find  the  measure  of  the  volume  of  a  cylinder^  take  the 
product  of  the  measures  of  the  area  of  the  base  and  the  altitude. 

V  =  bh. 

251.  The  curved  surface  of  a  right  circular  cone  can  be 
unwrapped  into  a  portion  of  a  circle.  Similarly  the  lateral 
surface  of  a  regular  pyramid  can  be  unwrapped  into  series 
of  equal  triangles  whose  common  altitude  is  equal  to  the 
slant  height  of  the  pyramid. 


Hence  the  measure  of  the  lateral  surface  of  a  right  circular 
cone  or  regular  pyramid  is  one-half  the  product  of  the  measure 
of  the  perimeter  of  its  base  by  that  of  its  slant  height. 

a  =3  \ps. 


MENSURATION 


311 


252.  Make  a  cylinder  out  of  paper  and  also  a  right  circular 
cone  having  the  same  altitude  and  base.  Fill  the  cone  with 
some  dry  material  and  empty  it  into  the  cylinder.  Do  this 
three  times  and  the  cylinder  will  be  just  tilled.  Hence  the 
volume  of  a  right  circular  cone  is  one-third  that  of  a  cylinder 
of  equal  base  and  altitude.  Hence^  to  find  the  volume  of  a 
right  circular  cone,  multiply/  one-third  the  measure  of  the  area 
of  the  base  by  the  measure  of  the  altitude. 

V  =z  ibh. 

Use  the  same  rule  to  find  the  volume  of  a  pyramid. 


253.  (1)  The  length  of  the  radius  of  the  base  of  a  right 
circular  cylinder  is  5  in.  and  its  altitude  is  8  in.  Find  its 
volume  and  the  area  of  its  curved  surface. 

The  measure  of  the  area  of  the  base  =  3.1416  x  25  =  78.54. 
The  measure  of  the  volume  of  the  cy Under  =:  8  x  78.54  =  628.32. 
.-.  the  volume  =  628.32  cu.  in. 
Again, 
■  The  measure  of  the  circumference  of  the  base  =  3.1416  x  10  =  31.416. 
The  measure  of  the  area  of  the  curved  surface  =  8  x  31.416  =  251.328. 

.-.  the  area  =  251.328  sq.  in. 
What  must  we  add  to  this  to  find  the  total  area  of  the  cylinder  ? 

(2)  Find  the  area  of  the  curved  surface  and  also  the 
volume  of  a  cone  whose  altitude  is  8  in.  and  whose  base  is 
12  in.  in  diameter. 

A 


312  ARITHMETIC 

Since  the  altitude  AC  ia  perpendicular  to  the  diameter  BCD,  the  triangle 
ACB  is  a  right  triangle  whose  sides  are  8  in.,  6  in.,  and  10  in. 
Hence  AB  is  10  in.    • 

Also  the  circumference  of  the  base  =  3.1416  x  12  in.  =  37.6992  in. 
The  measure  of  the  area  z=z  ]  x  ^  x  37.6992  =  188.496. 

.-.  the  area  =  188.496  sq.  in. 

What  must  we  add  to  this  to  find  the  total  area  of  the  cone  ? 

Again, 

The  altitude  of  the  cone  =  8  in. 

The  area  of  the  base  =  3.1416  x  62  or  113.0976  sq.  in. 

The  measure  of  the  volume  =  i  x  8  x  113.0976  =  301.5936. 

.'.  the  volume  =  301.59  cu.  in. 

Exercise  186 

1.  Find  the  area  of  the  curved  surface  of  a  right  circular 
cylinder,  the  circumference  of  whose  base  is  12  in.  and  altitude 
8  in. 

2.  Find  the  area  of  the  curved  surface  of  each  of  these  right 
circular  cylinders : 

(a)   Circumference  of  base  =  10  in.,  15  in.,  18  in. 
Altitude  =    9  in.,     8  in.,  12  in. 
(6)   Diameter  of  base  =  6  in.,  16  in.,  7  in. 
Altitude  =  4  in.,  14  in.,  5  in. 

3.  Find  the  area  of :  (1)  The  curved  surface  of  a  right  cir- 
cular cylinder  of  altitude  12  in.  the  diameter  of  whose  base  is 
14  in.     (2)  Both  ends.     (3)  The  total  area.     (Use  ^-.) 

4.  Find  the  total  area  of  a  right  circular  cylinder  the  radius 
of  whose  base  is  5  in.  and  altitude  8  in. 

5.  Find  the  volume  of  a  cylinder  the  area  of  whose  base  is 
18  sq.  in.  and  altitude  8  in. 

6.  Find  the  volume  of  each  of  these  right  circular  cylinders : 
(a)   Area  of  base  =  16  sq.  in.,  27  sq.  in.,  135  sq.  in. 

Altitude  =    5  in.,  9  in.,  15  in. 


MENSURATION  313 

(p)   Eadius  of  base  =  10  in.,  7  in.,  9  in.,  8  in. 
Altitude  =  10  in.,  4  in.,  6  in.,  8  in. 

7.  Find  the  area  of  the  lateral  surface  of  a  right  circular  cone 
the  circumference  of  whose  base  is  14  in.  and  slant  height  11  in. 

8.  Find  the  area  of  the  lateral  surface  of  each  of  these  right 
circular  cones: 

(a)  Circumference  of  base  =  26  in.,  17  in.,  1  ft.  6  in. 

Slant  height  =  20  in.,     8  in.,  9  in. 

(b)  Radius  of  base  =  5  in.,     8  in.,  i  yd. 

Slant  height  =  6  in.,  11  in.,  i  yd. 

9.  Find  the  area  of :  (1)  The  lateral  surface  of  a  right  cir- 
cular cone  the  radius  of  whose  base  is  6  in.  and  slant  height  8  in. 
(2)  The  area  of  its  base.     (3)  Its  total  area. 

10.  Find  the  total  area  of  the  surface  of  a  right  circular  cone 
the  radius  of  whose  base  is  5  in.  and  slant  height  12  in. 

11.  Find  the  area  of  the  lateral  surface  of  a  regular  pyramid 
whose  slant  height  is  12  in.  and  base  is  an  equilateral  triangle, 
one  of  whose  sides  is  8  in.  long. 

12.  Find  the  area  of  the  convex  surface  of  a  regular  pyramid 
whose  slant  height  is  8  in.  and  whose  base  is  a  square,  its  side 
being  4  in. 

13.  Find  the  total  area  of  the  pyramid  in  the  preceding 
example. 

14.  Find  the  volume  of  a  right  circular  cone  the  area  of  whose 
base  is  16  sq.  in.  and  altitude  6  in. 

15.  Find  the  volume  of  each  of  these  right  circular  cones  : 
(a)   Area  of  base  =  24  sq.  in.,  62  sq.  in.,  79  sq.  in. 

Altitude  =    7  in.,         15  in.,         14  in. 

(6)   Radius  of  base  =  6  in.,  9  in.,  10  in.,  15  in. 

Altitude  =  8  in.,  7  in.,  14  in.,  25  in. 


314  ARITHMETIC 

16.  Find  the  volume  of  a  square  pyramid  wliose  altitude  is 
18  in.  and  the  edge  of  whose  base  is  6  in. 

17.  Find  the  volume  of  each  of  these  square  pyramids : 

Altitude  =  16  in.,  12  in.,  25  in.,  2  ft.  6  in. 
Side  of  square  =    9  in.,     8  in.,  16  in.,  1  ft.  2  in. 

18.  Find  the  volume  of  a  pyramid  whose  altitude  is  15  in. 
and  base  an  equilateral  triangle  of  side  10  in. 

19.  The  radius  of  the  base  of  a  right  circular  cone  is  6  in.  and 
its  altitude  8  in.     Find  the  area  of  its  cone  surface. 

20.  The  radius  of  the  base  of  a  right  circular  cone  is  8  in.  and 
its  slant  height  17  in.     Find  its  volume. 

21.  Find  the  volume  of  a  pyramid  whose  altitude  is  15  in.  and 
base  (1)  a  rectangle  whose  sides  are  7|  in.  and  6  in. ;  (2)  a  triangle 
whose  sides  are  10  in.,  24  in.,  and  26  in.  long. 


254.  That  portion  of  a  cone  included  between  its  base 
and  a  plane  parallel  to  its  base  is  called  a  frustum  of  the 
cone. 

That  portion  of  a  pyramid  between  its  base  and  a  plane 
parallel  to  its  base  is  called  a  frustum  of  the  pyramid. 

255.  To  find  the  lateral  area  of  the  frustum  of  a  regular 
pyramid  or  of  a  right  circular  cone,  multiply  one-half  the  meas- 
ure of  its  slant  height  hy  that  of  the  sum  of  the  perimeters  of  its 
bases, 

a  =  l(p+p')s. 

To  find  the  volume  of  a  frustum  of  a  regular  pyramid  or  of  a 
right  circular  cone^  multiply  one-third  the  measure  of  its  alti- 


MENSURATION  315 

tude  hy  the  sum  of  the  areas  of  the  two  bases  and  the  square 
root  of  their  product, 

o 

Exercise  187 

1.  Find  the  area  of  the  lateral  surface  of  the  frustum  of  a 
right  circular  cone,  the  perimeters  of  whose  bases  are  27  and  23  in., 
and  slant  height  15  in. 

2.  Find  the  area  of  the  lateral  surfaces  of  each  of  these  frus- 
tums of  pyramids : 

p  =  12  in.,    25  in.,    1  ft.  10  in.,    2  ft.  3  in. 

p'  =  18  in.,    16  in.,    1  ft.    3  in.,    1  ft.  7  in. 

8=    S  in.,    10  in.,    1  ft.    2  in.,    2  ft.  1  in. 

3.  Find  the  lateral  area  of  a  frustum  of  a  right  circular,  cone 
whos^  slant  height  is  12  in.  and  whose  bases  are  circles  whose 
radii  are  6  in.  and  4  in. 

4.  Find  the  entire  area  of  the  frustum  in  example  3. 

5.  Find  the  lateral  area  of  the  frustum  of  a  regular  pyramid 
whose  slant  height  is  10  in.  and  bases  equilateral  triangles  of  side 
8  in.  and  6  in. 

6.  Find  the  entire  area  of  the  frustum  in  example  5. 

7.  Find  the  volume  of  the  frustum  of  a  pyramid  whose  bases 
contain  2  sq.  ft.  and  8  sq.  ft.  and  whose  altitude  is  6  ft. 

8.  Find  the  volume  of  the  frustum  of  a  square  pyramid,  the 
sides  of  whose  bases  are  4  ft.  and  5  ft.  and  altitude  12  ft. 

9.  Find  the  volume  of  a  frustum  of  a  cone  whose  bases  con- 
tain 27  sq.  in.  and  12  sq.  in.  and  whose  altitude  is  15  in. 

10.  Find  the  volume  of  the  frustum  of  a  right  circular  cone, 
the  radii  of  whose  bases  are  10  ft.  and  20  ft.  and  whose  altitude 
is  24  ft. 


316  ARITHMETIC 

11.  Find  the  volume  of  the  frustum  of  a  triangular  pyramid, 
the  sides  of  whose  bases  are  3  ft.,  4  ft.,  5  ft.,  and  6  ft.,  8  ft.,  10  ft., 
and  whose  altitude  is  7  ft. 

12.  Find  the  number  of  cubic  feet  in  a  log,  the  radii  of  whose 
ends  are  8  in.  and  1  ft.  and  length  30  ft. 

256.  The  measure  of  the  area  of  the  surface  of  a  sphere 

is  equal  to  four  times  the  square  of  the  radius  multiplied 

by  3.1416. 

a  =  4  X  3.1416  r^, 

257.  If  we  imagine  a  sphere  to  be  divided  into  a  large 
number  of  small  cones,  as  in  §  246  we  divided  the  circle 
into  triangles,  the  centre  of  the  sphere  being  the  vertex  of 
each  cone,  and  a  small  portion  of  the  circumference  being 
its  base,  we  can  think  of  the  volume  of  the  sphere  as  being 
equal  to  the  sum  of  the  volumes  of  the  cones.  The  altitude 
of  each  cone  is  equal  to  the  radius  of  the  sphere,  and  the 
total  area  of  their  bases  is  equal  to  the  area  of  its  surface. 
Hence  the  volume  of  the  sphere  is  given  by  the  formula ; 

F=  J r(4x  3.1416  x  r2) 
=  1  X  3.1416  ?'3. 

Hence  the  measure  of  the  volume  of  the  sphere  is  ^  of  3.1416 
times  the  cube  of  the  measure  of  the  radius, 

258.  (1)  Find  the  surface  of  a  sphere  whose  radius  is  6  in. 

The  measure  of  the  area  =  4  x  3.1416  x  6^  =  452.3904. 
.*.  the  area  =  462.39  sq.  in. 

(2)  Find  the  volume  of  a  sphere  whose  diameter  is  8  in. 
The  measure  of  the  volume  =  |  x  3.1416  x  4»  =  268.0832. 
.*.  the  volume  =  268.08  cu.  in. 


MENSURATION  317 

Exercise  188 

1.  Find  the  surface  of  a  sphere  whose  radius  is  3  in. 

2.  Find  the  surface  of  a  sphere  12  in.  in  diameter. 

3.  Find  the  volumes  of  the  spheres  given  in  examples  1  and  2. 

4.  Find  the  surface  of  a  sphere  5  ft.  in  diameter. 

5.  Find  the  volume  of  a  sphere  whose  diameter  is  16  ft. 

6.  Place  a  croquet  or  base  ball  between  two  chalk  boxes. 
Place  a  foot  measure  in  line  with  one  edge  of  each  box.  What 
is  the  diameter  of  the  ball  ?  What  is  the  area  of  its  surface  ? 
What  is  its  volume  ? 

7.  With  a  pair  of  compasses  draw  a  circle  with  the  diameter 
found  in  example  6.  Cut  out  this  circle  and  pass  the  ball  through 
the  hole. 

8.  If  the  pressure  of  the  air  is  equal  to  15  lb.  a  square  inch, 
what  is  the  pressure  on  the  surface  of  a  sphere  6  in.  in  diameter  ? 

Exercise  189 

1.  Find  the  cost  of  a  field  25  rd.  long  and  20  rd.  wide  at  $  96 
an  acre. 

2.  What  is  the  area  of  a  parallelogram  7  ft.  6  in.  long  and 

3  ft.  4  in.  wide  ? 

3.  The  base  of  a  triangle  is  15  ft.  9  in.  and  the  altitude  12 
ft.  4  in.     Find  its  area. 

4.  Find  the  number  of  square  yards  in  a  triangle  whose  sides 
are  13  ft.,  14  ft.,  15  ft. 

5.  The  perimeter  of  a  flower  bed  in  the  form  of  an  equilateral 

triangle  is  27  ft.     What  is  its  area  ? 

6.  How  many  acres  are  there  in  a  square  field  each  side  of 
which  is  330  yd.  ? 

7.  What  is  the  area  of  a  trapezoid  whose  parallel  sides  are 

4  ft.  6  in.  and  8  ft.  3  in.,  and  whose  altitude  is  5  ft.  3  in.  ? 


318  ARITHMETIC 

8.  What  is  the  value  of  a  field  in  the  form  of  a  trapezoid 
whose  parallel  sides  are  8.6  ch.  and  4.4  ch.,  and  whose  altitude  is 
5.4  ch,,  at  $  75  an  acre  ? 

9.  A  cow  is  fastened  to  a  stake  by  a  chain  40  yd.  long.  How 
many  square  yards  of  grass  more  than  an  acre  can  she  feed  on  ? 

10.  What  is  the  side  of  a  square  equal  in  area  to  a  circle  whose 
radius  is  100  ft.  ? 

11.  The  sides  of  a  triangular  wheat  field  are  275  yd.,  220  yd., 
165  yd.  Find  the  value  of  wheat  grown  on  it,  the  crop  averaging 
25  bu.  to  the  acre  and  worth  80  ^  per  bushel. 

12.  The  slant  height  of  a  triangular  pyramid  is  10  ft.  and  each 
side  of  the  base  2  ft.     Find  the  total  area  of  its  surface. 

13.  What  is  the  total  area  of  the  surface  of  a  right  cone  whose 
slant  height  is  20  in.  and  whose  base  is  a  circle  of  radius  10  in.  ? 

14.  What  is  the  area  of  the  lateral  area  of  a  right  cone  whose 
altitude  is  15  in.  and  the  diameter  of  whose  base  is  16  in.? 

15.  Find  the  total  area  of  the  surface  of  the  frustum  of  a 
square  pyramid,  each  side  of  the  basesj  being,  respectively,  10  in. 
and  6  in.,  and  the  slant  height  15  in. 

16.  Find  the  total  area  of  the  surface  of  a  frustum  of  a  cone, 
whose  greater  diameter  is  18  in.  and  less  diameter  8  in.,  and  slant 
height  24  in. 

17.  Find  the  area  of  the  surface  of  a  sphere  whose  diameter 
is  22  in. 

18.  Find  the  area  of  the  surface  of  the  earth,  supposing  it  a 
sphere  of  diameter  7960  mi. 

19.  Find  the  entire  surface  of  a  cylinder  whose  height  is  10 
ft.  and  base  a  circle  of  5  ft.  diameter. 

20.  Find  the  volume  of  a  square  prism  whose  length  is  5  ft. 
6  in.,  each  side  of  its  base  being  1  ft.  4  in. 

21.  How  many  gallons  of  water  will  a  prismatic  vessel  con- 
tain, its  base  being  a  rectangle  14  in.  by  6  in.,  and  its  altitude 
22  in.? 


MENSURATION  319 

22.  The  length,  of  a  cylindrical  piece  of  timber  is  18  ft.  and  the 
diameter  of  its  base  is  1  ft.     Find  its  volume. 

23.  Find  the  volume  of  a  square  pyramid,  each  side  of  whose 
base  is  4  ft.  and  height  12  ft. 

24.  A  conical  church  spire  is  100  ft.  high  and  the  diameter  of 
its  base  is  18  ft.     Find  its  volume. 

25.  What  is  the  volume  of  a  frustum  of  a  square  pyramid 
whose  height  is  6  ft.,  the  sides  of  the  greater  end  being  7  in.  and 
of  the  smaller  5  in.  ? 

26.  Find  the  volume  of  a  squared  piece  of  timber,  its  length 
being  18  ft.,  each  side  of  the  greater  end  being  18  in.  and  of 
the  small  end  12  in. 

27.  Find  the  volume  of  a  tapering  round  piece  of  timber  whose 
length  is  10  ft.  and  the  diameters  of  the  ends  8  in.  and  4  in. 
respectively. 

28.  Find  the  volume  of  a  sphere  2  ft.  in  diameter. 

29.  A  vessel  in  the  form  of  a  right  circular  cone  is  4  ft.  deep, 
and  the  diameter  of  its  base  is  3  ft.  Find  how  many  gallons  of 
water  it  will  contain. 

30.  A  cylindrical  cistern,  8  ft.  in  diameter  and  6  ft.  deep,  is  | 
full  of  water.     Find  the  number  of  gallons  of  water  in  the  cistern. 

31.  An  excavation  2  yd.  deep,  in  the  form  of  the  frustum  of  a 
square  pyramid,  has  its  upper  base  8  yd.  long,  and  its  lower  base 
4  yd.  long.  Find  the  number  of  wagon  loads  of  earth  required  to 
fill  it. 


CHAPTER   XIX 

THE  METRIC   SYSTEM   OF  "WEIGHTS   AND   MEASURES 

259.  The  French  or  Metric  System  of  Weights  and  Meas- 
ures is  based  upon  the  decimal  system.  It  is  used  in  scien- 
tific treatises,  and  has  been  adopted  by  most  of  the  nations  of 
Europe  and  South  America.  It  is  also  in  partial  use  in  the 
United  States  and  Canada. 

260.  The  fundamental  unit  of  the  metric  system  is  the 
Meter,  which  is  39.37  in.  long.  The  original  standard  meter 
is  a  platinum  rod,  called  the  French  Standard  Meter,  which 
is  deposited  in  the  Archives  at  Paris. 


,  1 

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l|ll||ll| 

iUiliU 

liii  hill 

lllllllll 

lllllllll 

lllllllll 

1111  nil 

lllllllll 

The  length  of  the  measure  in  the  diagram  is  -^  of  a  meter, 
and  is  called  a  decimeter.  It  is  divided  into  10  equal  parts, 
each  of  which  is  called  a  centimeter.  Each  of  these  is  divided 
into  10  equal  parts,  each  of  which  is  called  a  millimeter. 

261.  In  order  that  pupils  may  study  this  system  of  weights  and  measures 
to  the  best  advantage,  the  school  should  be  provided  with  a  system  of  metric 
weights  and  measures,  and  each  pupil  with  a  foot  rule  on  which  the  decimeter, 
centimeter,  and  millimeter  are  marked. 

An  intelligent  study  of  the  system  can,  however,  be  made  if  the  teacher 
has  a  metric  stick  and  a  liter  for  reference. 

820 


METRIC   SYSTEM  321 

262.  The  names  of  the  higher  or  lower  units  in  the  metric 
system  are  formed  by  attaching  certain  prefixes  to  the  names 
of  the  standard  units,  thus  : 

Beca  signifies  10  times  the  unit. 
Hecto  signifies  100  times  the  unit. 
Kilo    signifies  1000  times  the  unit. 

Bed  signifies  the  10th  part  or  .1  of  the  unit. 
Centi  signifies  the  100th  part  or  .01  of  the  unit. 
Milli  signifies  the  1000th  part  or  .001  of  the  unit. 

Units  of  Length 
1  millimeter  (mm.)    =  .001  meter 
1  centimeter  (cm.)     =  .01  meter 
1  decimeter  (dm.)      =  .1  meter 
1  meter  (m.)  =  standard  unit 

1  decameter  (Dm.)    =  10  meters 
1  hectometer  (Hm.)  =  100  meters 
1  kilometer  (Km.)      =  1000  meters 
1  myriameter  (Mm.)=  10,000  meters 

The  units  in  common  use  are  the  millimeter^  centimeter^  meter,  and  kilo- 
meter. 

263.  Write  out  the  table  of  the  units  of  length,  thus : 

(a)  10  millimeters  =  1    centimeter,  etc. 
(J)     1  millimeter    =  J^  centimeter,  etc. 

Exercise  190 

1.  Measure  with  a  yardstick  in  the  school  yard  a  distance  of 
11  yd.  Measure  along  the  same  distance  10  times  with  the  meter. 
What  is  the  difference  in  inches  ? 


322  ARITHMETIC 

2.  Reduce  11  yd.  and  also  10  m.  to  inches,  and  verify  the 
result  obtained  in  the  first  question. 

3.  Measure  the  length  of  the  schoolroom  in  meters  and  deci- 
mals of  a  meter. 

4.  Find  the  hypotenuse  of  a  right  triangle  whose  sides  are 
(1)  6  m.  and  8  m.,  (2)  24  cm.  and  45  cm. 

5.  Librarians  frequently  use  the  centimeter  as  the  unit  in 
registering  the  heights  of  books.  Express  in  centimeters  the 
height  of  your  (a)  Arithmetic,  (6)  History,  (c)  Geography,  (d)  and 
also  of  several  other  books. 

6.  Measure  and  express  in  terms  of  the  centimeter  the  dis- 
tances between  the  lines  on  ruled  paper. 

7.  Measure  and  express  in  terms  of  the  centimeter  the  height 
of  the  schoolroom  thermometer. 

8.  How  many  centimeters  are  there  in  a  full  line  of  this  book  ? 

9.  Press  tightly  together  the  leaves  of  this  book.  Make  a 
layer  just  1  cm.  thick.  Count  the  leaves  and  find  the  thickness 
of  one  leaf  as  a  decimal  of  a  millimeter. 

10.  Cut  a  slit  2  mm.  wide^  by  3  cm.  long,  in  a  sheet  of  paper. 

11.  How  many  millimeters  are  there  in  the  width  of  your 
pencil  ? 

12.  Find  the  number  of  inches  in  1  Km.,  reduce  the  result  to  a 
decimal  of  a  mile,  and  show  that  1  Km.  is  nearly  equal  to  |  of  a 
mile. 

13.  If  a  train  travels  at  the  rate  of  20  m.  a  second,  what  is  the 
rate  in  kilometers  per  hour  ? 

14.  Show  that  5  in.  are  very  nearly  equal  to  127  mm. 

15.  For  what  do  we  use  the  inch,  the  yard,  and  the  mile  ?  For 
what  are  the  millimeter,  the  centimeter,  the  meter,  and  the  kilo- 
meter respectively  used  ? 


UmnTS   OF   AREA  323 

UNITS  OF   AREA 

264.  The  principal  units  for  measuring  land  are  the  square 
meter,  called  the  Centare  (ca.),  the  square  decameter  called  the 
Are  (a.),  and  the  square  hectometer  called  the  Hectare  (ha.). 


1  sq. 

cm. 

1  sq.  dm. 

Each  side  of  this  square  measures 

1  dm.,  or 

3f|  in.,  very  nearly. 

A  liter  is  a  cube  each  face  of  which  has  the  dimensions  of  this 

square. 

A  gram  is  the  weight  of  a  cubic  centimeter  (see  small  square 

above)  of  distilled  water,  weighed  in  vacuo  at  temperature  of 

maximum  density,  39.1  F.      A  liter  or  cubic  decimeter  of  such 

water  weighs  1  kg.  or  2^  lb.  nearly. 

265.  How  many  units  of  length  are  there  in  the  side  of  a 
square  meter,  the  decimeter  being  the  unit  ? 

How  many  units  of  area  are  there  in  a  square  meter,  the 
square  decimeter  being  the  unit  ? 


324  ARITHMETIC 

How  many  units  of  area  are  there  in  a  square  decimeter, 
the  square  centimeter  being  the  unit  of  area  ? 

What  part  of  a  square  meter  is  a  square  decimeter  ? 

Write  a  square  decimeter  as  a  decimal  of  a  square  meter. 

Write  a  square  centimeter  as  a  decimal  of  a  square  deci- 
meter. As  a  decimal  of  a  square  meter.  What  is  the  ratio 
of  each  unit  of  area  in  the  metric  system  to  the  next  smaller, 
and  also  to  the  next  higher  ? 

Unit8   of  Area 

1  square  millimeter  (q.  mm.)=  .000001  of  a  square  meter 
I  1  square  centimeter  (q.  cm.)    =  .0001  of  a  square  meter 

;  1  square  decimeter  =  .01  of  a  square  meter 

I  1  square  meter  (q.  m.)  =  standard  unit 

\  1  square  decameter  =  100  square  meters 

J  1  square  hektometer  =  10,000  square  meters 

1  square  kilometer  (q.  Km.)    =  1,000,000  square  meters 

1  centare .{Q,2i..)  =  .01  are  (a.) 

1  hectare  (ha.)  =  100  ares 

Note  that  the  square  meter  is  1^  times  as  large  as  the  square  yard. 

Exercise  191 

1.  Cut  out  of  paper  1  q.  dm.  and  also  1  q.  cm.  What  is  the 
ratio  of  the  two  areas  ? 

2.  Draw  1  sq.  ft.  and  measure  it  with  1  q.  dm.  of  paper  as  the 
unit  of  area. 

3.  A  square  decimeter  contains  .10764  sq.  ft.  Find  the  num- 
ber of  square  decimeters  contained  in  a  square  foot,  correct  to  two 
decimal  places,  and  compare  the  result  with  that  obtained  in 
example  2. 

4.  Make  a  drawing  of  1  q.  m.,  and  draw  1  sq.  yd.  within  it. 

5.  Mark  out  an  are  on  the  school  ground. 

6.  State  the  table  of  area,  expressing  each  imit  of  area  as  equal 
to  100  times  the  next  lower. 

7.  Find  the  area  of  a  page  of  this  book  in  square  centimeters. 


UNITS  OF  VOLUME  325 

UNITS   OF  VOLUME 

266.  The  principal  units  of  volume  are  the  cubic  meter, 
also  called  the  Stere,  and  the  cubic  decimeter,  called  the  Liter. 

267.  How  many  units  of  length  are  there  in  a  side  of  a 
meter,  the  decimeter  being  the  unit  ? 

How  many  units  of  volume  are  there  in  the  cubic  meter  or 
stere,  the  cubic  decimeter  or  liter  being  the  unit  ? 

What  is  the  ratio  of  each  unit  of  volume  in  the  metric 
system  to  the  next  smaller  unit  ?     To  the  next  larger  ? 

Units  of  Volume 
1  cubic  millimeter  (c.  mm.)=  .000000001  of  a  cubic  meter 
1  cubic  centimeter  (c.  cm.)  =  .000001  of  a  cubic  meter 
1  cubic  decimeter  =  .001  of  a  cubic  meter 

1  cubic  meter  (c.  m.)  =  staridard  unit 

Exercise  192 

1.  Make  a  liter  out  of  paper. 

2.  Fill  a  quart,  liquid  measure,  with  sand  and  empty  it  into 
your  liter.     Which  of  the  two  measures  is  larger  ? 

3.  Fill  a  liter  with  sand  and  empty  it  into  a  quart,  dry  meas- 
ure.    Which  of  the  two  is  larger  ? 

4.  Fill  a  gallon  measure,  using  the  liter  as  a  dipper,  and  note 
how  many  liters  are  equivalent  to  the  gallon. 

5.  1  liter  is  equal  to  .264  gal.;  find,  correct  to  two  decimal 
places,  the  number  of  liters  in  a  gallon.  Compare  this  result  with 
that  obtained  in  example  4. 

6.  State  some  purposes  for  which  the  liter  is  used. 

7.  Make  a  cubic  centimeter  out  of  paper. 

8.  Make  the  necessary  measurements  and  compute  the  volume 
of  the  room  in  cubic  centimeters. 

9.  Make  the  necessary  measurements  and  compute  the  volume 
of  a  box  in  liters. 


326  AiirniMP:Tic 

10.  Express  as  a  decimal  part  of  a  cubic  meter  the  volume  of  a 
beam  3  m.  long,  10  cm.  wide,  and  5  cm.  thick. 

11.  A  cylindrical  vessel  having  a  base  of  a  square  meter  is 
filled  with  water  to  the  depth  of  2  m.  How  many  liters  of  water 
does  it  contain  ? 

12.  How  many  liters  of  water  may  be  held  by  a  vessel  meas- 
uring 25  X  35  X  75  cm.  ? 

13.  What  will  it  cost  to  build  a  wall  1  Hm.  long,  J  dm.  thick, 
and  1  m.  high,  at  $  5  a  cubic  meter  ? 

UNITS   OF  WEIGHT 

268.  The  principal  units  of  weight  are  the  Gram  and  the 
Kilogram. 

The  Gram  is  the  weight  of  a  cubic  centimeter  of  distilled 
water  at  40°,  at  which  temperature  water  is  at  its  maximum 
density. 

A  nickel  weighs  5  g. 

A  liter  of  distilled  water  at  40°  weighs  1  kg.  The  kilogram  is  nearly  equal 
to  2^  lb.  Avoirdupois. 

A  cubic  meter  of  water  at  40°  weighs  a  metric  ton  (1000  kg.). 

Bxeroise  103 

1.  One  gram  is  equal  to  15.432  gr.  Show  that  1  kg.  is 
approximately  equal  to  2^  lb.  Avoirdupois. 

2.  A  cubic  meter  of  distilled  water  at  40°  weighs  how  many 
kilograms  ?  If  1  kg.  is  equal  to  2i  lb.  Avoirdupois,  how  many 
pounds  does  1  c.  m.  of  water  weigh  ? 

3.  Show  that  a  metric  ton  weighs  about  10%  more  than  our 
short  ton. 

4.  If  sulphuric  acid  is  1.8  times  as  heavy  as  water,  what 
weight  of  the  acid  will  a  two-liter  bottle  contain? 

5.  What  part  of  a  liter  is  750  g.  of  water  ? 


UNITS   OF   WEIGHT  327 

6.  What  is  the  weight  of  1  deciliter  of  water  ? 

7.  If  alcohol  is  80%  as  heavy  as  water,  what  will  375  c.  cm. 
of  alcohol  weigh  ? 

8.  If  20  c.  cm.  of  lead  weighs  227  g.,  what  is  the  ratio  of  the 
weight  of  lead  to  that  of  an  equal  volume  of  water  ? 

9.  If  a  quantity  of  iron  weighs  7.8  times  as  much  as  an  equal 
quantity  of  water,  what  is  the  weight  of  an  iron  bar  75  x  4  x  3  cm.  ? 

10.  A  body  weighing  512  g.  in  air  weighs  428  g.  in  water. 
What  per  cent  of  its  weight  is  lost  ? 

11.  A  liter  flask  was  two-fifths  filled  with  water;  the  remain- 
ing space  being  filled  with  sand,  the  weight  was  found  to  be 
2050  g.     Eequired  the  weight  of  a  liter  of  sand. 

12.  If  the  pressure  of  the  air  on  the  surface  of  a  body  is  1  kg. 
to  the  square  centimeter,  what  is  the  pressure  of  the  air  on  the 
surface  of  a  sphere  whose  radius  is  10  cm.  ? 

13.  A  cubical  block  of  ice  measures  3  dm.  along  its  edge. 
What  will  be  its  weight  if  ice  weighs  94%  as  much  as  an  equal 
volume  of  water  ? 

14.  What  is  the  weight  of  air  in  a  room  5  m.  long,  3  m.  wide, 
4  m.  high,  if  1  c.  dm.  of  air  weighs  .0018  kg.  ? 

15.  Emperor  William  of  Germany  was  struck  in  the  face  with 
a  fishplate  weighing  550  g.     Find  its  weight  in  pounds. 


CHAPTER  XX 
Miscellaneous  Exercise  194 

1.  Write  the  following  in  figures: 

(a)  Fifty  thousand  nine  hundred  nine. 

(6)  Nine  hundred  thousand  ninety. 

(c)  Six  hundred  fifty  thousand  seven  hundred. 

(d)  Eight  hundred  seven  thousand  eight. 

(e)  Seven  hundred  seventy  thousand  sixty-seven. 
(/)  Nine  million  ninety  thousand  ninety-nine. 
(g)  Eighty  million  nine  hundred  thousand  thirty. 

(h)  Nine  hundred  seventy  million  eight  hundred  eighty-seven 
thousand. 

(i)    Six  hundred  seventeen  million  ninety-three. 

(j)   Nine  hundred  nineteen  thousand  four  hundred  eleven. 

(k)   Six  hundred  four  thousand  twenty-five. 

2.  Write  in  figures : 

Twenty-five  thousand  four  hundred  ninety;  ninety-nine  thou- 
sand nine  hundred  seventeen ;  nine  hundred  seven  thousand  six 
hundred  six  ;  one  million ;  MDCCCXCV.  And  in  words  :  9009  j 
16,060;  7018;  207,509;  75,115. 

3.  (a)  Define  and  give  examples  of  quantity,  unit,  and  number. 
(6)  Explain  the  basis  of  our  system  of  numeration. 

4.  Write  in  figures  (placed  for  addition):  Nine  hundred  nine- 
teen; three  hundred  eleven;  seven  hundred  seventy;  eight  hun- 
dred ninety-seven;  six  hundred  eight;  three  hundred  nine;  XCVII; 
LXVII;  CXIX;  CDL;  and  DCXL. 

328 


MISCELLANEOUS  EXERCISE 


329 


5.  Add:  4567890123 

5678901234 
6789012345 
7890123456 
8901234567 
0912345678 
6598695326 
8396876549 
7788995566 
3453453456 

6.  Write  down  neatly  the  following  statement  of  six  weeks' 
cash  receipts;  add  the  amounts  vertically  and  horizontally,  and 
prove  the  correctness  of  the  work  by  adding  your  results : 


MON. 

TUES. 

Wed, 

Thur. 

Fri. 

Sat. 

Total 

1st 

$28.79 

$34.71 

$35.33 

$30.10 

$27.97 

$47.81 

2d 

23.87 

30.03 

29.38 

33.84 

26.77 

48.77 

3d 

16.99 

27.09 

28.77 

30.16 

24.95 

43.07 

4tli 

29.13 

33.72 

30.81 

39.17 

28.47 

50.05 

5th 

18.47 

32.29 

26.73 

34.45 

28.88 

54.39 

6th 

19.02 

27.06 

29.04 

29.89 

29.51 

61.93 

Total 

7. 

Solve,  as 

5  in  exan 

aple  6 : 

MON. 

TUES. 

Wed. 

Thue. 

Fri. 

Sat. 

Total 

1st 

$65.95 

$24.89 

$79.79 

$40.78 

$37.59 

$89.61 

2d 

58.71 

41.65 

24.67 

94.26 

70.26 

42.51 

3d 

47.58 

99.57 

50.60 

80.71 

91.82 

89.76 

4th 

29.69 

70.80 

87.91 

74.93 

36.63 

21.90 

■ 

5th 

81.45 

56.93 

54.82 

96.57 

12.72 

96.67 

6th 

42.63 

68.77 

81.79 

60.86 

31.87 

75.82 

Total 

330  ARITHMETIC 

8.  Mr.  Jones  bought  one  house  for  $865  and  another  for 
$984,  and  sold  them  both  for  $  1900.     How  much  did  he  gain  ? 

9.  Juliette  has  $149,  Florence  has  $87  more  than  Juliette, 
and  Elizabeth  has  $  115  more  than  both.  How  many  dollars  has 
Elizabeth  ? 

10.  Thanksgiving  Day,  1901,  the  Michigan  University  football 
team  increased  its  record  of  451  points  by  50.  Find  its  record 
for  the  season  of  1901. 

11.  Two  men  together  receive  $97.75,  but  one  receives  $18.25 
more  than  the  other.     How  much  does  each  receive  ? 

12.  A  and  B  start  together  and  walk  in  the  same  direction,  A 
at  the  rate  of  4  mi.  an  hour,  and  B  at  the  rate  of  3  mi.  an  hour. 
At  the  end  of  7  hr.  A  turns  and  goes  back.  How  many  miles  will 
B  have  gone  when  he  meets  A  ? 

13.  In  a  factory,  12  men,  16  women,  and  30  boys  are  employed. 
At  the  end  of  a  week  they  receive  $  330.  A  man  is  paid  as  much 
as  2  women,  and  a  woman  as  much  as  3  boys.  What  is  the  share 
of  each  ? 

14.  A  man  bought  a  number  of  cows  for  $1080;  he  sold  half 
of  them  for  $  810,  thereby  gaining  $  15  on  each  one  sold.  What 
did  each  cow  cost  ? 

15.  A  clerk  received  a  salary  of  $650  a  year.  He  spent  50  f^ 
a  day  the  first  year,  $4  a  week  the  second  year,  and  $22  a 
month  the  third  year.     How  much  did  he  save  in  three  years  ? 

16.  The  subtrahend  is  9564,  the  remainder  is  1965.  What  is 
the  minuend  ?  The  multiplier  is  96  and  the  product  is  82,848. 
What  is  the  multiplicand  ? 

17.  The  dividend  is  1800,  the  quotient  is  17,  and  the  remainder 
66.     What  is  the  divisor  ? 

18.  How  many  times  can  506  be  subtracted  from  the  product 
of  6072  and  13,986  ? 

19.  The  quotient  of  a  division  is  834.  What  quotient  would 
have  been  obtained  if  both  dividend  and  divisor  had  been  first 
Ittultiplied  by  13  ?     Why  ? 


MISCELLANEOUS   EXERCISE  Ml 

20.  Subtract  847^J  from  1003 3%,  explaining  fully  each  step. 

21.  Simplify  i-  — |of  f +7-^?  ^^^  ^^^  ^^^  many  times  the 
result  is  contained  in  f  -^  (-^  of  y\  —  ^). 

22.  Divide  the  Sum  of  f  of  8^  and  2|  of  5|  by  the  difference 
between  f  of  3^  and  ^  of  -|-  of  2|. 

23.  Prove,  (1)  |  of  |  =  y\;  (2)  |  of  |  =  |  of  |. 

24.  Simplify  31+ ?i^-:^  of  3i--. 

25.  A  boy's  age  now  is  ^  of  his  father's.  In  6  yr.  it  will  be 
■|  his  father's  present  age.     How  old  is  he  ? 

26.  A  house  and  lot  are  together  worth  $  2100 ;  J  of  the  value 
of  the  house  is  equal  to  ^  of  the  value  of  the  lot.  Find  the  value 
of  each. 

27.  The  circumference  of  a  wheel  is  ^-  of  its  diameter.  Find 
the  diameter  of  a  wagon  wheel  which  makes  360  revolutions  in 
going  a  mile. 

28.  A  man  owned  a  f-interest  in  a  mill,  and  sold  ^  of  his 
interest  to  one  man,  and  -^  of  his  interest  to  another.  What  part 
of  the  mill  did  each  of  the  three  men  then  own  ? 

29.  If  to  a  certain  number  its  |,  J,  and  i  be  added,  the  sum 
will  be  122 ;  required  the  number. 

30.  Find  the  number  which  is  207  more  than  the  sum  of  -J-  and 
^  of  itself. 

31.  A  man  spent  -^  of  his  money  for  a  house,  f  of  the  remain- 
der for  cattle,  and  the  rest  for  a  farm.  If  the  farm  cost  him 
$  357  less  than  the  house  and  cattle  together,  what  did  he  pay 
for  all  ? 

32.  A  legacy  of  ^9500  is  to  be  divided  among  A,  B,  and  C, 
so  that  A  will  get  y\  of  the  whole,  and  B  will  get  f  as  much  as 
C.     Find  the  shares  of  each. 

33.  A  man  spent  ^  of  his  money  for  provisions,  f  of  the 
remainder  for  clothing,  -f^  of  the  remainder  for  charity,  and  had 
$  9.10  left.     How  much  did  he  have  at  first  ? 


332  ARITHMETIC 

34.   John  Smith  sells  a  merchant  752  lb.  of  cheese  at  11 J  ^ 
per  pound,  and  receives  the  following  goods  in  exchange : 


11yd.  silk  @$  2.25 
4001b.  sugar  @4J^ 
12  lb.  raisins  @  11  ^ 

3  pr.  gloves  @  75  ^. 
Find  the  balance  due  John  Smith. 


961b.  nails  @3|^; 

56  yd.  gray  cotton  @  9|  ^ ; 

11  yd.  white  cotton  @  10^; 


35.  A  man  owns  a  horse  and  saddle;  J  of  the  value  of  the 
horse  is  equal  to  4  times  the  value  of  the  saddle ;  the  horse  and 
saddle  together  are  worth  $  170.     Find  the  value  of  each. 

36.  A  man  bought  a  horse  and  carriage  for  $  280,  and  |  of  the 
cost  of  the  carriage  was  equal  to  J  of  the  cost  of  the  horse.  What 
was  the  cost  of  each  ? 

37.  Divide  the  product  of  .037  and  .0025  by  the  sum  of  .9,  .02, 
and  .005. 

38.  Divide  6  by  .000725,  correct  to  four  decimal  places. 

39.  Add  together  1.302,  3.2589,  and  40.93.  Multiply  the  sum 
by  .00297  and  divide  the  product  by  90.09. 

40.  Multiply  350.4  by  .0105  and  divide  the  product  by  .0000219. 

41.  What  decimal  must  be  taken  from  the  sum  of  69J,  8.2, 
5.445,  .065,  and  20j^,  so  that  it  will  contain  6.05  an  exact  number 
of  times  ? 

42.  A  drover  lost  .065  of  his  flock  by  wolves,  .105  by  disease, 
and  .27  by  theft.  He  then  sold  .75  of  what  remained,  and  had 
280  sheep  left.     Find  the  number  in  his  original  flock. 

43.  Find  the  amount  of  the  following  bill : 

1328  ft.  siding,  at  $  1.62J  per  C. ; 
48,480  cu.  ft.  timber,  at  f  59.37^  per  M. ; 
7400  fence  rails,  at  $  7.75  per  C. ; 
8400  fence  pickets,  at  $  15.00  per  M.; 
5680  lb.  hay,  at  ^  12.50  per  T. 


MISCELLANEOUS  EXERCISE  333 

44.  A  cooper  paid  ^78.32  for  16,488  bbl.  staves.  Kequired 
the  price  per  M. 

45.  A  rectangular  field  is  7  ch.  75  1.  long  and  4  ch.  87^  1.  wide. 
How  many  rods  of  fencing  are  required  to  enclose  it  ? 

46.  How  many  miles  of  road,  3  rd.  wide,  will  contain  8  A.  of 
land? 

47.  Make  a  drawing  that  will  show  the  number  of  square  yards 
in  a  square  rod.     (Scale  1  yd.  to  1  in.) 

48.  Find  the  value  of  a  piece  of  land  20  ft.  x  40  rd.,  at  $  1000 
per  acre. 

49.  A  certain  map  is  drawn  on  a  scale  of  8  mi.  to  an  inch.  On 
this  map  the  township  of  Scott  measures  ly\  in.  in  length  and 

11  in.  in  width.     How  many  acres  does  it  contain  ? 

50.  Find  the  expense  of  sodding  a  plot  of  ground  which  is 
40  yd.  long  and  100  ft.  wide,  with  sods  each  1  yd.  in  length  and 
1  ft.  in  breadth,  the  sods,  when  laid,  costing  75^  per  hundred. 

51.  A  floor  16  ft.  8  in.  by  14  ft.  2  in.  is  to  be  laid  with  square 
tiles.  Find  the  dimensions  of  the  largest  tiles  that  can  be  used 
without  cutting  or  fitting. 

52.  Find  the  cost  of  papering  a  room  24  ft.  long,  21  ft.  wide, 

12  ft.  high,  at  25^  a  roll,  12  yd.  long  and  21  in.  wide. 

53.  How  much  will  it  cost  to  plaster  the  walls  and  ceiling 
of  a  room  15  ft.  long,  12  ft.  wide,  and  11  ft.  high,  at  32^^  per 
square  yard? 

54.  A  room  18  ft.  by  16  ft.  is  carpeted  with  carpet  |  yd.  wide, 
and  the  smallest  possible  number  of  yards  of  the  carpet  is  used. 
Find  (a)  the  number  of  breadths,  (6)  the  number  of  yards. 

55.  How  many  thousand  shingles,  18  in.  long  and  4  in.  wide, 
lying  ^  to  the  weather,  are  required  to  shingle  the  roof  of  a 
building  54  ft.  long,  with  rafters  22  ft.  long,  the  first  row  of 
shingles  being  double? 


334  ARITHMETIC 

56.  A  schoolroom  is  30  ft.  long,  24  ft.  wide,  and  10  ft.  high 
above  the  wainscoting.  The  trustees  pay  $  20  per  thousand  for 
a  new  floor,  ^15  per  thousand  for  a  new  board  ceiling,  10^  per 
square  yard  for  painting  the  ceiling,  4  ^  per  square  yard  for  tint- 
ing the  walls,  and  $  2  per  day  for  6  da.  labor.  Find  the  total 
cost. 

57.  A  cubical  cistern  is  5  ft.  deep.  How  many  gallons  of  water 
will  it  hold  when  it  is  |  full  ? 

58.  How  many  cubical  blocks,  each  edge  of  which  is  ^  ft.,  are 
equivalent  to  a  block  of  wood  8  ft.  long,  4  ft.  wide,  and  2  ft. 
thick? 

59.  If  the  ceiling  of  a  square  room  is  15  ft.  high,  how  many 
square  feet  of  floor  must  it  have  in  order  that  50  pupils  and  the 
teacher  may  each  have  300  cu.  ft.  of  air  ? 

60.  Four-foot  wood  piled  5|-  ft.  high  requires  how  many  feet  in 
length  of  the  pile-  for  2^  cd.  ? 

61.  What  is  the  value  of  a  pile  of  wood  360  ft.  long,  12  ft. 
wide,  6  ft.  high,  at  |  3.20  per  cord  ? 

62.  A  square  plot  of  ground  that  contains  -^^  A.  is  covered 
with  cordwood  (4  ft.  long)  to  an  average  height  of  12  ft.  What 
is  the  wood  worth  at  $  4.12  per  cord? 

63.  Required  the  cost  of  35  pieces  of  scantling  18  ft.  long, 
4  in.  wide,  and  2  in.  thick,  at  $  14  per  thousand,  board  measure. 

64.  How  many  board  feet  are  there  in  12  scantlings  16  ft.  by 
4  in.  by  2  in.  ? 

65.  It  is  required  to  build  a  sidewalk  J  mi.  in  length,  8  ft.  wide, 
and  2  in.  thick,  supported  by  three  continuous  lines  of  scantlings 
4  in.  square.     What  will  the  lumber  cost  at  $  17  per  thousand  ? 

66.  Find  the  value  of  the  following  lumber  at  $  15  per 
thousand :  —         20  pieces  2  x     4,  18  ft.  long ; 

20  pieces  4  x     4,  12  ft.  long ; 
20  pieces  3  x  10, 16  ft.  long. 


MISCELLANEOUS  EXERCISE  335 

67.  A  farmer  sold  a  lot  of  barley,  weighing  2712  lb.,  when 
barley  was  40^  per  bushel.  In  weighing  the  grain,  the  dealer 
made  a  mistake  and  took  it  as  rye,  and  paid  for  it  at  49^  per 
bushel.     How  much  did  the  farmer  gain  or  lose  by  the  mistake  ? 

68.  The  weight  of  a  cubic  foot  of  water  is  621  Jb.^  and  1  gal. 
contains  231  cu.  in.     Find  the  weight  in  ounces  of  1  pt.  of  water. 

69.  A  lake  whose  area  is  45  A.  is  covered  with  ice  3  in.  thick. 
Find  the  weight  of  the  ice  in  tons,  if  1  cu.  ft.  weighs  920  oz.  Avoir. 

70.  In  what  time  would  a  field,  80  by  60  rd.  pay  for  under- 
draining  lengthwise,  at  2)^  per  foot,  if  the  field  yields  2  bu.,  at  66  j^, 
per  acre  more  than  before  draining  ?  The  drains  are  4  rd.  apart, 
and  the  first  drain  runs  down  the  centre  of  the  field. 

71.  Find  the  amount  of  the  following  bill: 

June  1,  1896,  G.  Murray  &  Co.  sold  to  John  Scott,  4886  bu. 
36  lb.  wheat  @  58  ^  per  bushel,  4532  lb.  peas  @  52  ^  per  bushel, 
38  bu.  3  pk.  barley  @  54^  per  bushel,  465  lb.  flour  @  $  1.50  per 
hundredweight,  4685  lb.  bran  @  ^  15  per  ton.  Write  out  a  receipt 
in  full  for  payment  of  account,  June  26. 

72.  Find  the  length  of  the  shortest  line  that  can  be  exactly 
measured  by  a  yard  measure,  a  ten-foot  pole,  or  a  two-rod  chain. 

73.  Required  the  cost  of  1  doz.  silver  spoons,  each  weighing 
18  pwt.  18  gr.  at  ^1.15  per  ounce. 

74.  Reduce  7  gal.  3  qt.  1  pt.  to  the  fraction  of  a  barrel. 

75.  I  sow  11  bu.  2  pk.  4  qt.  of  wheat,  and  raise  therefrom 
215  bu.  2  qt.     How  much  is  the  average  yield  per  bushel  of  seed  ? 

76.  The  running  time  of  the  Empire  State  Express  from  New 
York  to  Buffalo  is  8  hr.  30  min.,  and  the  distance  is  440  mi.  If 
stops  of  5  min.  each  are  made  at  Albany,  Utica,  Syracuse,  and 
Rochester,  what  is  its  average  speed  per  hour  ? 

77.  Find  (a)  the  exact  number  of  days  from  Jan.  17,  1899,  to 
April  5,  1899 ;  (b)  the  difference  in  time  by  subtraction  of  dates. 

78.  A  railroad  train  moves  1  mi.  in  65  sec.  What  is  its  speed 
per  hour  ? 


336  ARITHMETIC 

79.  A  Dote  given  August  15,  1901,  for  90  days,  will  mature 
when? 

80.  A  can  walk  3J  mi.  in  50  min.,  and  B  can  walk  2J  mi.  in 
36  min.  How  many  yards  will  A  be  ahead  of  B  when  A  has 
gone  6  mi.,  if  they  start  together  ? 

81.  A  farmer  delivered  at  a  warehouse  four  loads  of  wheat 
weighing  respectively  2113  lb.,  2310  lb.,  2270  lb.,  and  2091  lb. 
How  much  should  he  have  received  at  72^  per  bushel? 

82.  The  difference  in  longitude  between  two  places  being 
9°  34'  25",  what  is  the  difference  in  time? 

83.  A  man  has  a  salary  of  $400  a  year,  and  has  $500  in  the 
bank.  If  he  spends  $  500  a  year,  in  what  time  will  his  money  be 
all  gone  ?     (Allow  no  interest.) 

84.  What  is  the  shortest  stick  that  can  be  cut  into  pieces, 
9  in.,  12  in.,  or  15  in.  in  length,  with  nothing  remaining  ? 

85.  (a)  What  is  meant  by  a  Common  Multiple  of  two  or  more 
numbers  ? 

(6)   Find  the  L.  C.  M.  of  36,  54,  105. 

86.  (a)  What  is  meant  by  the  prime  factors  of  a  number  ? 
(6)  Find  the  prime  factors  of  13,230,  22,050,  and  23,625. 

(c)  By  means  of  the  prime  factors  find  their  G.  C.  M.  and 
L.  C.  M. 

87.  Resolve  16,335  and  18,018  into  their  prime  factors,  and 
from  inspection  of  these  write  the  prime  factors  of  their  (a) 
L.  C.  M.  and  (b)  G.  C.  M. 

88.  A  farmer  bought  a  number  of  horses  and  cows  for  $2000. 
There  were  3  times  as  many  cows  as  horses,  and  a  horse  costs 
twice  as  much  as  a  cow.  If  each  horse  costs  $80,  how  many 
cows  did  he  buy  ? 

89.  The  difference  in  weight  of  two  chests  of  tea  is  25  lb. ;  the 
value  of  both  at  65^  per  pound  is  $113.75.  How  many  pounds 
of  tea  are  in  each  chest  ? 


MISCELLANEOUS  EXERCISE  337 

90.  What  is  the  smallest  sum  of  money  with  which  you  can 
buy  chickens  at  25  i^,  or  geese  at  50  ^,  or  turkeys  at  75  ^,  or  lambs 
at  $  3,  or  sheep  at  $  5,  or  pigs  at  $  7,  or  cows  at  $  35,  or  horses 
at  $  140,  and  have  exactly  $  15  left  for  expenses  ? 

91.  Ten  cents  will  buy  3  oranges,  4  lemons,  or  5  apples.  How 
many  apples  are  worth  as  much  as  5  doz.  oranges  and  7  doz. 
lemons  ? 

92.  One  workman  charges  $  3  for  a  day's  work  of  8  hr.,  and 
another  $3.50  for  a  day's  work  of  9  hr.  Which  had  I  better 
employ,  and  how  much  shall  I  have  to  pay  him  for  work  that  he 
can  do  in  a  fortnight,  working  6  hr.  a  day  ? 

93.  A  can  do  a  piece  of  work  in  |  of  a  day,  and  B  in  i  of  a 
day.     If  $  1.40  is  paid  for  the  work,  how  much  should  A  receive  ? 

94.  How  many  oranges  must  a  boy  buy  and  sell  to  make  a 
profit  of  $9.30,  if  he  buys  at  the  rate  of  5  for  3  ^,  and  sells  at  the 
rate  of  4  for  3^? 

95.  A  and  B  dig  a  ditch  in  50  hr.  With  C's  help  they  could 
have  done  it  in  18J  hr.  In  what  time  could  C  do  f  of  the  work 
alone  ? 

96.  Three  men  can  dig  a  certain  drain  in  8  da.  They  work  at 
it  for  5  da.,  when  one  of  them  falls  ill,  and  the  other  two  finish 
the  work  in  5  da.  more.  How  much  of  the  work  did  the  first  man 
do  before  he  fell  ill  ? 

97.  A  boy  can  run  6  times  around  a  circular  plot  of  ground  in 
52  sec. ;  another  boy  can  run  9  times  around  the  same  plot  in  80 
sec.  If  they  start  from  the  same  place  at  the  same  time,  and  run 
in  the  same  direction,  how  many  rounds  will  each  make  before 
the  faster  boy  overtakes  the  slower  ? 

98.  Express  in  the  form  of  a  vulgar  fraction  the  average  of 
I  A,  .7,  .4^,  and  .4861. 

99.  In  a  granary  there  are  4  bins,  each  10  ft.  long  and  5  ft. 
wide.  How  high  must  they  be  boarded  in  front  to  be  capable  of 
holding  860  bushels  ? 


338  ARITHMETIC 

100.  The  outfit  of  a  livery  stable  is  worth  $  3000 ;  ^  the  value 
of  the  horses  is  equal  to  ^  the  value  of  vehicles,  harness,  etc. 
Find  the  value  of  the  horses. 

101.  A  farmer  agreed  to  pay  his  hired  man  10  sheep  and  $  160 
for  1  yr.  labor.  The  man  quit  work  at  the  end  of  7  mo,,  receiving 
the  sheep  and  $  60  as  a  fair  settlemei^t.  Find  the  value  of  each 
sheep. 

102.  Divide  $  1200  among  A,  B,  and  C,  so  that  A  may  have 
$  70  more  than  B,  and  twice  as  much  as  C. 

103.  A  train  going  25  mi.  an  hour  starts  at  1  o'clock  p.m.  on  a 
trip  of  280  mi. ;  another  going  37  mi.  an  hour  starts  for  the  same 
place  at  12  min.  past  4  o'clock  p.m.  When  and  where  will  the 
former  be  overtaken  ? 

104.  In  the  number,  28,672,  the  value  expressed  by  the  first 
two  digits  from  the  left  is  how  many  times  the  value  expressed 
by  the  fourth  digit  from  the  left  ? 

105.  A  town  whose  population  was  10,000  increased  10% 
every  year  for  3  yr.  What  was  the  population  at  the  end  of 
that  period  ? 

106.  A  house  and  lot  was  sold  for  $  7030,  at  a  loss  of  16|  % 
of  its  cost.     Find  the  cost. 

107.  Five  men  in  a  factory  accomplish  as  much  as  8  boys. 
What  per  cent  of  a  man's  work  does  a  boy  do  ?  What  per  cent 
of  a  boy's  work  does  a  man  do  ? 

108.  Forty-five  per  cent  of  a  carload  of  melons  were  sold  to 
one  dealer,  and  33^%  of  those  left  to  another.  How  many  were 
there  in  the  car  before  any  were  sold,  if  after  the  second  sale 

P' there  remained  110? 

109.  In  a  certain  school  48%  of  the  pupils  are  boys,  and  there 
are  39  girls.     Find  the  number  of  boys. 

110.  How  many  pounds  of  flour  will  be  required  to  make  1000 
lb.  of  bread,  if  the  bread  weigh  30%  more  than  the  flour  used? 

111.  93  lb.  6  oz.  is  what  per  cent  of  43  lb.  12  oz.  ? 


MISCELLANEOUS   EXERCISE  339 

112.  Water  in  freezing  expands  10%.  If  1  cu.  ft.  of  water 
weighs  1000  oz.,  find  the  weight  of  1  cu.  ft.  of  ice. 

113.  Give  answers  to  the  following: 

(a)  15f  %  of  660  =  ?  {d)  .2%  of  40  =  ? 

(5)   660  is  15|%  of  what  number  ?     (e)  40  is  .2%  of  what  number? 

(c)  f  is  what  per  cent  of  f  ? 

(/)  What  per  cent  of  itself  must  be  added  to  a  number  so  that 
the  sum  diminished  by  10%  of  itself  may  be  17%  more  than  the 
original  number  ? 

114.  Brooms  are  bought  wholesale  at  $20  a  gros^.  What  per 
cent  profit  will  be  made  by  selling  them  at  20  ^  each  ? 

115.  A  merchant  purchases  sugar  at  $4.50  per  hundredweight. 
At  what  price  per  pound  must  he  sell  it  in  order  to  gain  5f  %? 

116.  I  bought  a  house  for  $4000  and  spent  40%  of  the  cost 
in  repairs.  What  must  I  rent  it  for  a  month  in  order  to  make 
a  clear  gain  of  5%  of  the  total  cost,  taxes  and  repairs  amounting 
to  $  80  yearly  ? 

117.  By  selling  a  piano  for  $260,  a  dealer  loses  20%.  How 
much  should  he  have  sold  it  for  to  gain  5  %  ? 

118.  A  man  having  lost  20%  of  his  capital  is  worth  exactly 
as  much  as  another  who  has  just  gained  15%  on  his  capital. 
The  second  man's  capital  was  originally  $  9000.  What  was  the 
first  man's  capital  ? 

119.  A  dealer  sold  an  article  for  $8.10  and  lost  10%.  At 
what  selling  price  would  he  have  gained  10%? 

120.  A  bookseller  deducts  10%  from  the  market  price  of  his 
books,  and  after  this  has  a  gain  of  25%.  He  sells  a  book  for 
$7.20.  Find  the  cost  price  of  the  book,  and  what  per  cent  the 
marked  price  is  in  advance  of  the  cost  price. 

121.  A  merchant  bought  1000  yd.  of  carpet  at  60/  a  yard,  and 
sold  I  of  it  at  a  profit  of  30%,  i  at  a  profit  of  20%,  and  the  rest 
at  a  loss  of  20%.     How  much  did  he  receive  for  the  carpet  ? 


340  ARITHMETIC 

122.  A  sells  goods  to  B  at  a  gain  of  12%,  and  B  sells  the  same 
goods  to  C  at  a  gain  of  7^%.  C  paid  $3762.50  for  the  goods. 
How  much  did  A  pay  for  them  ? 

123.  A  machinist  sold  two  seed-drills  for  equal  sums  of  money. 
He  gained  25  %  on  the  one  and  lost  25  %  on  the  other.  His  total 
loss  was  $  9.60.     Find  the  cost  of  each  drill. 

124.  H  purchased  a  house  and  lot  for  $3300,  paid  $1325  for 
repairs,  and  now  rents  the  premises  for  $30  a  month.  If  he 
expends  annually  for  taxes  $47.50,  and  for  incidental  repairs 
$  35,  what  is  his  per  cent  of  annual  income  on  his  investment  ? 

125.  A  merchant  closed  out  a  stock  of  cloaks  for  $311.04, 
at  a  loss  of  28^.     Kequired  the  loss  by  the  transaction. 

126.  By  selling  my  cloth  for  $  1.26  per  yard  I  gain  11  ^  more 
than  I  lose  by  selling  it  at  $  1.05  per  yard.  What  would  I  gain 
by  selling  800  yd.  at  $1.40  per  yard  ? 

127.  A  merchant  marks  his  goods  at  40%  in  advance  of  cost, 
and  in  selling  uses  a  pound  weight  \  oz.  too  light.  If  he  throws 
off  10%  of  his  marked  price,  find  his  gain  per  cent. 

128.  State  the  relation  between  1  lb.  Troy  and  1  lb.  Avoirdu- 
pois. What  is  the  gain  per  cent  when  the  selling  price  per  ounce 
Avoirdupois  is  the  same  as  the  cost  per  ounce  Troy  ? 

129.  A  man  bought  a  bankrupt  stock  at  60^  on  the  dollar  of 
the  invoice  price,  which  was  $4840.  He  sold  half  of  it  at  10% 
advance  on  invoice  price,  half  the  remainder  at  20%  below  invoice 
price,  and  the  balance  at  50%  of  invoice  price.  His  expenses 
were  10%  of  his  investment.  Find  his  loss  or  gain  (a)  in  money, 
and  (b)  in  rate  per  cent. 

130.  The  list  price  of  an  article  is  $150.  If  trade  discounts 
of  25%  and  16  J  %  are  allowed,  what  is  the  net  price? 

131.  A  dealer  buys  stoves  at  a  discount  of  22%  from  list  price, 
and  sells  them  at  list  price ;  what  is  his  per  cent  of  gross  profit 
on  the  investment  ? 


MISCELLANEOUS   EXERCISE  341 

132.  Kequired  the  net  price  of  an  article  listed  at  $400,  30%, 
10%,  and  5%  off. 

133.  From  the  list  price  of  a  line  of  goods  a  purchaser  is 
allowed  a  trade  discount  of  20%  ;  a  further  discount  of  10% 
off  the  trade  price  for  taking  a  quantity,  and  a  still  further  dis- 
count of  5%  off  his  bill  for  cash.  Find  his  gain  per  cent  by  sell- 
ing at  10%  less  than  the  list  price. 

134.  The  net  price  of  a  reaper  is  $158.40,  and  the  trade  dis- 
counts allowed  are  20%  and  10%.     Find  the  list  price. 

135.  A  commission  merchant  sold  coffee  for  me  and  remitted 
$1960,  after  deducting  his  commission  of  2%.  What  is  the 
value  of  the  coffee  ? 

136.  If  an  agent  receives  $  1092  to  buy  pork,  how  many  pounds, 
at  6^^  per  pound,  can  he  buy  and  retain  his  commission  of  5% 
for  buying  ? 

137.  A  commission  merchant  sold  1014  bu.  of  oats,  at  41^  per 
bushel,  paid  $33.74  freight  charges,  and  retained  3i%  commis- 
sion.    How  much  should  he  remit  to  the  consignor  ? 

138.  A  lad  earned  $21.16  collecting  accounts  for  a  physician. 
He  was  allowed  5f%.     What  amount  did  he  collect? 

139.  Find  the  premium  paid  to  insure  a  house  worth  $  7500 
for  I  of  its  value  for  3  yr.,  the  rate  for  each  year  being  -1%  of  the 
policy. 

140.  What  premium  must  be  paid  to  insure  a  cargo  of  4880  bu. 
of  wheat,  valued  at  78^  per  bushel,  at  1-|%,  the  policy  being  for 
only  I  of  its  value  ? 

141.  A  building  is  insured  for  $400  more  than  |  of  its  cost  at 
4%.  If  destroyed,  the  loss  will  be  $216.  Find  the  cost  of  the 
building. 

142.  A  dealer  shipped  200  bbl.  of  apples  to  Liverpool;  the 
average  cost  of  the  apples  was  $  3.75  per  barrel.  For  what  sum 
must  he  have  the  apples  insured  at  |%  premium  to  guard  against 
all  loss,  in  case  of  shipwreck,  his  other  expenses  being  $  75  ? 


342  ARITHMETIC 

143.  If  in  a  certain  town  ^3093.75  was  raised  from  a  |%  tax, 
what  was  the  assessed  valuation  of  the  property  in  the  town  ? 

144.  A  tax  of  $  24,750  is  levied  on  a  town,  the  assessed  valua- 
tion being  15  mills  on  a  dollar.  What  tax  does  a  man  pay  on  an 
income  of  $1100,  of  which  $400  is  exempted? 

145.  A  farmer  whose  property  is  assessed  at  $9600  pays  on 
the  dollar  If  mills  for  township  rates,  IJ  for  county  rates,  1^  for 
railway  bonus,  and  2^  for  school  rate.  How  much  does  he  pay 
in  all  ? 

146.  B's  tax  was  $86.2755  when  the  rate  was  7.635  mills  on  a 
dollar.     What  was  the  assessed  valuation  of  his  property  ? 

147.  A  certain  school  section  is  assessed  for  $150,000.  The 
trustees  have  built  a  schoolhouse  costing  $1800. 

(a)  What  will  the  schoolhouse  cost  a  ratepayer  whose  property 
is  assessed  for  $  4500  ? 

(6)  What  would  be  the  rate  of  taxation  per  annum  on  the 
whole  section  if  the  house  were  paid  for  in  six  equal  annual 
payments,  without  interest? 

148.  A  clerk  pays  $7.50  taxes  on  his  salary.  What  is  his 
total  salary  if  $  400  of  it  is  exempt  from  taxation  and  a  2^%  rate 
is  levied  on  the  remainder  ? 

149.  What  per  cent  must  be  assessed  on  $  1,500,000  to  produce 
$29,400  after  paying  2%  for  collecting? 

150.  An  importer  receives  an  invoice  of  kid  gloves  billed  at 
$680,  pays  a  duty  of  50%  ad  valorem,  and  sells  them  at  an 
advance  of  33^%  on  their  gross  cost  to  him.  How  does  the  price 
paid  by  the  purchaser  compare  with  the  exporter's  price  ? 

151.  A  merchant  imports  75  cases  of  indigo,  gross  weight  196 
lb.  each,  allowing  15%  for  tare.  What  was  the  duty  at  5^  per 
pound  ? 

152.  What  will  $1  amount  to  in  216  da.,  at  7J%  per  annum, 
simple  interest? 


MISCELLANEOUS  EXERCISE  343 

153.  Find  the  simple  interest  on  $  597.50  for  5  mo.  12  da.  at 
8%  per  annum. 

154.  How  long  will  it  take  $450,  at  S%,  to  yield  $21.30 
interest  ? 

155.  What  amount  will  be  due  July  1,  1902,  on  a  note  of  $  80, 
drawn  Feb.  6, 1901,  and  bearing  interest  at  5\%  per  annum,  exact 
interest  ? 

156.  Find  the  sum  due  Sept.  2,  1899,  on  a  note  for  $147.33, 
given  Jan.  13,  1899,  and  bearing  interest  at  4%  per  annum. 

157.  Find  the  exact  interest  on  $225  from  July  13,  1899,  to 
Sept.  3,  1899,  at  6%. 

158.  Find  the  interest  on  $  1,  at  7|-%  per  annum,  from  Jan.  1, 
1899,  to  June  3,  1899.     (Complete  answer  required.) 

159.  What  sum  will  amount  to  $354.09  in  7  mo.,  at  3%  per 
annum  ? 

160.  Ill  what  time  will  $1350  earn  $31.88  at  5%  per 
annum? 

161.  Find  the  face  of  a  draft  that  cost  $434.70,  at  |% 
premium. 

162.  If  the  interest  is  $12.57,  the  time  8  mo.  2  da.,  and  the 
rate  per  annum  5^%,  what  is  the  principal? 

163.  Find  the  exact  interest  on  $150  from  July  16  to  Dec.  9, 
at  5%  per  annum. 

164.  A  person  borrows  money  for  6  mo.  at  4%,  simple  interest, 
and  repays  at  the  end  of  the  time,  as  principal  and  interest,  $  816. 
How  much  did  he  borrow  ? 

165.  Find  the  simple  interest  on  $912.50,  at  8%,  from  Feb. 
13,  1901,  to  Dec.  19,  1902. 

166.  A  note  of  $360,  drawn  April  20,  1900,  was  paid  July  2, 
1901,  with  interest  at  7^%  per  annum.  Find  the  amount  paid, 
simple  interest. 


344  ARITHMETIC 

167.  Oct.  15,  1899,  a  young  man  deposited  in  the  savings 
bank  the  sum  of  $860.7o.  May  20,  1900,  he  withdrew  the  prin- 
cipal and  simple  interest  at  4%  per  annum.  What  amount  did 
he  withdraw  ? 

168.  Bought  a  horse  for  ^160,  and  gave  in  payment  my  note 
dated  Aug.  15,  1899,  with  interest  at  7|^%  per  annum  until  paid. 
Jan.  9,  1900,  I  sold  the  horse  for  $200  cash,  and  paid  my  note. 
What  was  my  net  gain  ? 

169.  If  for  $  7  I  can  have  the  use  of  f  35  for  3  yr.  4  mo.,  how 
much  a  month  shall  I  have  to  pay  for  the  use  of  $  8750  ? 

170.  Jan.  1,  1899,  a  person  borrowed  $2417.50  at  6|%,  sim- 
ple interest,  promising  to  return  it  as  soon  as  it  amounted  to 
$  2582.50.     On  what  day  did  the  loan  expire  ?     (365  da.  =  1  yr.) 

171.  March  1,  1899,  a  storekeeper  bought  goods  amounting,  at 
catalogue  prices,  to  $840,  on  which  he  was  allowed  successive 
discounts  of  33  J  %  and  5%.  The  account  is  payable  in  60  da., 
after  which  time  interest  is  to  be  charged  at  7%  per  annum. 
June  1,  1899,  he  paid  $  100.     How  much  is  due  July  1,  1899  ? 

172.  Pind  the  proceeds  of  a  note  for  $  200  given  at  Albany, 
N.Y.,  for  3  mo.,  and  discounted  at  the  bank  the  day  it  was  made 
at  6%. 

173.  Find  the  proceeds  of  a  note  for  $168  due  Oct.  20,  1899, 
and  discounted  Sept.  25,  1899,  at  a  Brooklyn,  N.Y.,  bank,  at  6% 
per  annum. 

174.  $  1234^%.  St.  Louis,  Jan.  16,  1899. 
Ninety  days  after  date,  I  promise  to  pay  A.  Bee,  or  order,  the 

sum  of  one  thousand  two  hundred  and  thirty-four  j^^*^  dollars,  at 
the  Bank  of  Commerce  here.     Value  received.  q    p^^. 

This  note  was  discounted  Feb.  10,  1899,  at  6%  per  annum. 
Find  the  proceeds. 

175.  A  note  for  $230,  drawn  Jan.  2,  1899,  at  3  mo.,  and  bear- 
ing interest  at  8%  per  annum,  was  discounted  Feb.  1  at  7%.  Find 
the  proceeds. 


MISCELLANEOUS  EXERCISE  345 

176.  Find  the  proceeds  of  the  following  note : 

$2400.  Hamilton,  Ohio,  Feb.  3,  1899. 

Five  months  after  date,  value  received,  I  promise  to  pay 
Thomas  Cowan,  or  order,  the  sum  of  two  thousand  four  hundred 
dollars,  at  the  Bank  of  Hamilton,  with  interest  at  6  %  per  annum. 

Vance  Allen. 
Discounted  May  22,  1899,  at  7%. 

177.  The  discount  on  a  note  for  $  3600,  which  matured  April 
21,  1899,  and  was  discounted  Feb.  24,  1899,  was  $  45.60.  Find 
the  rate  of  discount. 

178.  A  buys  600  yd.  of  silk  at  95^  per  yard,  and  sells  it  at  once, 
receiving  in  payment  a  90-day  note  for  $  700,  which  he  at  once 
discounts  at  a  bank  at  6%  per  annum.     Find  the  gain. 

179.  For  what  sum  must  a  note  be  drawn  June  1,  1899,  pay- 
able in  90  da.,  so  that  when  discounted  June  14,  at  8%,  the  pro- 
ceeds will  be  $717.20? 

180.  Jan.  1,  A  owes  a  bank  $  15,000.  He  offers  for  discount 
certain  notes:  $2500  due  Feb.  15,  $3700  due  March  13,  and 
$  7500  due  April  1.  If  these  are  discounted  at  8  %  per  annum, 
how  much  cash  must  he  pay  ? 

181.  Find  the  proceeds  of  a  note  for  $292.73,  discounted  at 
a  bank,  for  35  da.,  at  6%  per  annum,  exact  interest  method. 

182.  Find  the  value  of  (1.03)^ 

183.  A  man  has  the  choice  of  loaning  his  money  at  7^%,  com- 
pound interest,  or  at  8%,  simple  interest,  money  and  interest  to 
be  paid  at  end  of  3  yr.     Show  which  is  the  better  investment. 

184.  An  annual  deposit  of  $  250  is  made  with  a  loan  company 
which  pays  4%  per  annum  on  deposits,  compounded  half-yearly. 
Find  the  amount  of  all  these  deposits  when  the  fourth  has  been 
made. 

185.  June  30,  1899,  I  borrowed  $  16.50,  to  be  returned  April 
30,  1901.  With  compound  interest  at  6i%,  what  amount  did 
I  then  pay  ? 


346  ARITHMETIC 

186.  A  man  puts  $  350  in  a  savings  bank  each  year,  making 
his  first  deposit  Dec.  31,  1899.  How  much  will  there  be  to  his 
credit  Jan.  1,  1903,  the  bank  adding  4%  per  annum  ? 

187.  ^  1200  is  to  be  divided  between  two  persons,  A  and  B,  so 
that  A's  share  is  to  B's  share  as  2  to  7.     Find  the  share  of  each. 

188.  What  is  the  ratio  of  3J  to  |  ?     Answer  in  per  cent. 

189.  Divide  1026  into  four  parts  that  shall  be  in  the  ratio 
of  3,  11,  17,  and  23. 

190.  An  upright  pole  16  ft.  long  casts  a  shadow  5  ft.  4  in. 
long,  and  at  the  same  hour  the  shadow  of  a  tree  is  found  to 
be  26  ft.  9  in.     Required  the  height  of  the  tree. 

191.  The  sum  of  three  numbers  is  940.  The  first  number 
equals  f  of  the  second,  and  the  second  equals  -^^  of  the  third. 
Find  the  numbers. 

192.  One-sixth  of  the  square  of  a  certain  number  is  384.  Find 
the  number. 

193.  Find  the  square  root  of  .6  correct  to  three  decimal  places. 

194.  Find,  within  one  inch,  the  side  of  a  square  whose  area'  is 
5  A. 

195.  A  rectangular  field  whose  length  is  ^  of  its  width  contains 

2  A.  112  sq.  rd.     Find  the  length  of  a  diagonal. 

196.  Required  the  base  of  a  right-angled  triangle  whose  hypote- 
nuse is  16|-  ft.,  and  perpendicular  9}  ft. 

197.  A  ladder  78  ft.  long  stands  perpendicularly  against  a 
building.  How  far  must  it  be  pulled  out  at  the  foot  that  the  top 
may  be  lowered  6  ft.  ? 

198.  A  road  runs  round  a  circular  pond;  the  outer  circum- 
ference is  440  yd.,  and  the  width  of  the  road  is  20  yd.  Find  the 
area  of  the  pond. 

199.  In  order  to  drain  a  swamp  a  ditch  was  dug  1  mi.  long, 

3  ft.  deep,  6  ft.  wide,  at  the  surface,  and  4  ft.  wide  at  the  bottom. 
Find  the  total  cost  at  9  ^  per  cubic  yard. 

200.  How  many  gallons  of  water  will  a  circular  cistern  6  ft.  in 
diameter  and  7  ft.  deep  contain?     (1  cu.  ft.  =  7.48  gal.) 


MISCELLANEOUS  EXERCISE  347 

201.  The  surface  of  a  cube  is  432  sq.  ft.    What  is  its  volume  ? 

202.  (a)  A  circular  cistern,  8  ft.  in  diameter  and  9  ft.  in  depth, 
is  filled  with  water  to  the  height  of  6  ft.  How  many  gallons  of 
water  in  the  cistern  ? 

(b)  If  a  sphere  whose  diameter  is  4  ft.  is  submerged  in  the 
water  in  the  cistern,  how  high  will  it  cause  the  water  to  rise  ? 

203.  How  many  cords  are  there  in  a  cylindrical  log  20  ft.  long 
and  3  ft.  6  in.  in  diameter  ? 

204.  Find  the  diameter  of  a  circle  whose  area  is  equal  to  the 
sum  of  the  areas  of  two  circles  whose  diameters  are  12  in.  and 
16  in.  respectively. 

205.  Find  the  area  of  the  curved  surface  of  a  right  circular 
cone  the  radius  of  whose  base  is  3.5  in.  and  whose  altitude  is 
7  in. 

206.  A  chord  of  a  circle,  whose  radius  is  12  in.,  subtends  a 
right  angle  at  the  centre  of  the  circle.  Find  the  area  of  the 
smaller  segment  cut  off  by  this  chord. 

207.  A  spherical  shell,  internal  diameter  14  in.,  is  filled  with 
water.  Its  contents  are  poured  into  a  cylindrical  vessel  whose 
internal  radius  is  14  in.  Find  the  depth  of  the  water  in  the 
cylinder. 

208.  The  sides  of  a  triangle  are  40,  45,  and  50  ft.  respectively. 
Find  its  area. 

209.  The  diameter  of  a  circular  plate  of  lead  is  13  in.  From 
this  is  cut  out  a  circular  plate  of  radius  6  in.,  and  the  remainder 
of  the  lead  is  moulded  into  the  form  of  a  circular  plate,  with  \  of 
the  former  thickness.     Find  the  diameter  of  this  plate. 

210.  The  sides  of  a  triangle  are  13,  14,  and  15  ft.  Find  its 
area. 

211.  The  external  dimensions  of  a  rectangular  covered  box 
made  of  inch  staff,  are  7,  8,  and  9  ft.  Find  the  capacity  of  the 
box  and  the  quantity  of  lumber  in  it. 

212.  A  ball  of  yarn  3  in.  in  diameter  makes  one  mitten.  How 
many  similar  mittens  will  a  ball  6  in.  in  diameter  make  ? 


348  ARITHMETIC 

213.  Find  the  volume  of  a  cylinder  the  radius  of  whose  base  is 
10  in.,  the  altitude  being  18  in. 

214.  Find  the  volume  of  a  cone  the  radius  of  whose  base  is  10 
in.,  the  altitude  being  18  in. 

216.  How  often  can  the  cone  in  example  214  be  filled  and 
emptied  into  the  cylinder  in  example  213  ? 

216.  The  length  of  the  radius  of  the  base  of  a  right  circular 
cylinder  is  9  in.  and  its  altitude  is  16  in.     Find  the  volume. 

217.  Find  the  area  of  the  curved  surface  of  the  cylinder  in 
example  216.     Find  the  area  of  its  entire  surface. 

218.  Find  the  volume  of  a  cone  whose  altitude  is  15  in.  and 
whose  base  is  a  circle  10  in.  in  diameter. 

219.  Find  the  volume  of  a  cone  whose  altitude  is  12  in.  and 
the  diameter  of  whose  base  is  5  in. 

220.  Find  the  area  of  the  curved  surface  of  a  cone  whose  alti- 
tude is  20  in.  and  the  radius  of  whose  base  is  15  in.  Find  also 
its  total  area. 

221.  If  the  diameter  of  a  cylindrical  well  be  5  ft.,  and  its 
depth  27  ft.,  how  many  cubic  yards  of  earth  were  removed  in 
order  to  form  it? 

222.  A  farmer  employs  a  number  of  men  and  8  boys ;  he  pays 
the  boys  ^.65  and  the  men  $1.10  per  day.  The  amount  that  he 
paid  to  all  was  as  much  as  if  each  had  received  $.92  per  day. 
How  many  men  were  employed  ?     (x  men.) 

223.  Two  men  start  from  the  same  point  at  the  same  time  to 
walk  in  the  same  direction  around  a  block  of  land  IJ  mi.  on  each 
side.  A  goes  at  the  rate  of  4  mi.  and  B  3  mi.  an  hour.  How  far 
will  A  walk  before  he  overtakes  B  ? 

224.  Find  the  cost  of  the  material  required  to  fence  2^  mi.  of 
railway  (both  sides),  posts  placed  8  ft.  apart,  an  8-in.  base  1  in. 
thick,  a  2  X  4  in.  rail  at  top,  and  6  strands  of  wire.  The  posts 
cost  12^^  each,  the  lumber  $  14  per  thousand,  and  the  wire  4^  per 
pound.     (A  pound  of  wire  stretches  1  rd.) 


CHAPTER   XXI 

APPENDIX 
STOCKS   AND   BONDS 

269.  The  capital  of  a  bank  or  other  public  company  is 
called  Stock. 

It  is  usually  divided  into  a  definite  number  of  equal  parts 
or  Shares. 

The  original  value  of  a  share,  generally  $100,  f  50,  or  $25, 
is  called  its  Par  Value. 

270.  The  Market  Value  of  a  share  is  the  sum  for  which  it 
can  be  sold. 

Stock  is  said  to  be  above  par^  or  at  a  premium,  when  the 
market  value  is  greater  than  its  par  value  ;  it  is  said  to  be 
below  par^  or  at  a  discount,  when  the  market  value  of  the 
share  is  less  than  its  par  value. 

Thus  if  $100  stock  sells  for  $112  money,  the  stock  is  at 
12%  premium,  and  it  is  said  to  sell  at  112. 

If  $100  stock  sells  for  $96  money,  the  stock  is  at  4%  dis- 
count, and  is  quoted  at  96. 

271.  A  Stock  Broker  is  a  person  who  buys  or  sells  stocks, 
bonds,  or  similar  securities.  His  commission,  called  Brokerage, 
is  reckoned  at  a  certain  rate  per  cent,  which  varies,  the  most 
common  rate  being  ^  ot  1%  or  -|%. 

272.  A  Bond  is  a  note  bearing  interest  issued  by  a  govern- 
ment or  corporation.  There  are  two  kinds  of  bonds  — 
registered  and  coupon  bonds. 

349 


350 


ARITHMETIC 


A  Registered  Stock  or  Bond  is  one  which  is  registered  on 
the  books  of  the  company  or  government  issuing  it,  and 
which  cannot  be  sold  or  transferred  except  in  writing  at  the 
office  of  the  treasurer. 

An  Interest  Coupon  is  an  interest  certificate  payable  to  the 
bearer,  which  is  attached  to  the  bond,  and  which  is  detached 
when  the  interest  becomes  due. 

One  coupon  is  attached  to  the  bond  for  each  instalment 
of  interest  to  be  paid  on  it. 

273.  The  following  is  the  quotation  of  United  States 
bonds  in  the  market  of  Oct.  11,  1901 : 


Bid 

Asked 

Bid 

Asked 

New  2s  .    .    .    .    . 

109 

109^ 

Coupon  4s     .     .     . 

112 

112| 

Coupons     .... 

109 

109^ 

Registered  4s  new  . 

139 

140 

New  3s 

107J 

108i 

Coupon  4s  new  .     . 

139 

140 

New  3s  coupon  .     . 

108 

109 

Registered  5s     .     . 

107^ 

108i 

New  3s  small .    .    . 

108 

109 

Coupon  5s     .     .     . 

lOTi 

108J 

Registered  4s .    .     . 

112 

112| 

274.   The  following  is  the  quotation  of  stock  in  the  market 
of  Oct.  11,  1901 : 


Closing 

Stocks 

Highest 

Lowest 

Oct.  n 

Oct.  10 

Am.  Sugar 

llOf 

im 

119 

118. J 

Amal.  Copper 

90J 

88i 

89| 

89; 

Atchison 

79 

77f 

78i 

nit 

C.  R.  I.  &  P 

144 

140^ 

142i 

146 

Del.  &  Hudson 

16«i 

•     166 

166 

165| 

Manhattan 

12U 

119J 

121 

120J 

So.  Pacific 

59| 

67| 

69| 

67 

STOCKS  AND  BONDS  351 

Exercise  195 

1.  At  what  different  prices  is  Amalgamated  Copper  stock 
quoted,  Oct.  11,  1901  ? 

2.  What  will  a  seller  receive  from  his  broker  for  1  share  of 
Atchison  stock,  Oct.  11, 1901,  at  each  of  the  quoted  prices,  broker- 
age being  i%  ?     What  from  1  share  of  Am.  Sugar  ? 

3.  What  will  a  buyer  have  to  pay  for  1  share  of  Manhattan 
stock  at  each  quotation,  Oct.  11,  1901,  brokerage  i%?  What  for 
C.  E.  I.  &  F.  ? 

4.  At  what  per  cent  premium  are  the  different  quotations  for 
Am.  Sugar,  C.  R.  I.  &  P.,  and  Del.  &  Hudson  stock,  Oct.  11, 1901  ? 

5.  At  what  per  cent  discount  are  the  different  quotations  for 
Amal.  Copper,  Atchison,  and  So.  Pacific  stock,  Oct.  11,  1901  ? 

6.  What  would  I  receive  for  1  share  of  Del.  &  Hudson,  Oct. 
11,  1901,  sold  at  the  highest  price,  brokerage  i-%?  What  for  10 
shares  ?     What  for  100  shares  ? 

7.  What  would  I  have  to  pay  for  1  share  of  Atchison  stock, 
Oct.  11,  1901,  bought  at  the  lowest  price,  brokerage  \%"^  What 
for  10  shares  ?     What  for  100  shares  ? 

8.  What  would  I  have  to  pay  for  1  share  of  Am.  Sugar  stock, 
Oct.  11,  1901,  at  the  lowest  quoted  price,  brokerage  ^%?  What 
for  10  shares  ?     What  for  100  shares  ? 

9.  What  would  I  receive  for  1  share  of  So.  Pacific  stock,  Oct.' 
11,  1901,  sold  at  the  lowest  quotation,  brokerage  -g^%  ?  What  for 
10  shares  ?     What  for  100  shares  ? 

10.  What  will  1  share  of  Amal.  Copper  stock  cost,  Oct.  11, 1901, 
at  the  closing  price,  brokerage  |^%  ?  How  many  shares  can  I 
buy  for  %  180  ?     For  %  450  ?     For  ^  360  ? 

11.  What  will  1  share  of  Atchison  stock  cost,  Oct.  11,  1901,  at 
the  lowest  quotation,  brokerage  i%?  How  many  shares  can  be 
bought  for  %  155  ?     For  %  llh  ? 

12.  What  is  the  difference  between  the  highest  and  lowest 
quotations  of  Manhattan  stock,  Oct.  11,  1901  ? 


352  ARITHMETIC 

13.  What  is  the  difference  between  the  closing  prices  of  Del.  & 
Hudson  stock,  Oct.  10  and  Oct.  11,  1901  ? 

14.  What  is  the  difference  between  the  highest  and  lowest 
prices  of  Atchison  stock,  Oct.  11,  1901  ? 

15.  At  what  per  cent  discount  is  stock  which  is  quoted  at  88  ? 
98  J?     72  J? 

275.  (1)  How  much  will  be  realized  by  selling  out  66 
shares  of  Missouri  Pacific  stock  at  95 J,  brokerage  J%  ? 

1  share  of  stock  sells  for  $  95|  —  $  |,  or  $ 9of  money. 

.♦.  66  shares  of  stock  sell  for  66  x  $95f  or  $6294.75  money. 

Note.  —The  brokerage  =  66  x  $ |  =  $  8.25. 

(2)  How  many  shares  of  Manhattan  stock  at  124  J,  broker- 
age J%,  can  I  buy  for  $5591.25  ? 

1  share  costs  $  124^  +  $  ^,  or  $  124.25. 

/.  the  number  of  shares  =  $  5591.21  -=-  $  124.26  =  45. 

Note.  —  The  brokerage  =  45  x  $  ^  =  $  5.62^. 

(3)  A  broker  realizes  $7.25  from  a  sale  of  stock,  brokerage 
^fo'     What  was  the  par  value  of  the  stock  sold? 

I  %  of  the  par  value  =  #7.26. 
1  %  of  the  par  value  =  $  68. 
100  %  of  the  par  value  =  $  5800. 
.*.  the  par  value  =  $  5800. 

(4)  I  sold  through  my  broker  95  shares  of  Twin  City 
Rapid  Transit  stock,  receiving  for  it  $9476.25,  brokerage  ^%. 
Find  at  what  price  the  stock  was  quoted. 

95  shares  sell  for  $9476.25. 

1  share  sells  for  $  9476.26  -,-  96  =  $  99.75  ; 

i.e.  excluding  brokerage,  the  selling  price  =  99i. 

.-.  stock  is  quoted  at  99f  +  J  or  99f 


STOCKS  AND  BONDS  353 

(5)  "What  annual  income  will  be  realized  from  f  3828.12-|-, 
invested  in  the  U.  S.  3's  at  109J,  brokerage  l%? 

1  share  costs  $  lOQi  -f  ^  |  =  $  109f  =  .$  109.375. 
The  number  of  shares  =  $  3828.125  -~  $  109.375  =  35. 
.-.  the  income  =  35  x  $  3  =  $  105. 

(6)  What  amount  of  money  must  be  invested  in  6%  stock 
at  119|,  brokerage  |^%,  to  realize  an  income  of  1978  ? 

1  share  yields  an  income  of  $  6. 
The  number  of  shares  =  $ 978  h-  $6  =  163. 
1  share  costs  $  119|  +  $  i  =  $  119|. 
163  shares  cost  163  x  $  119|  =  $  19,539.62|. 
.-.  $  19,539. 62 1  must  be  invested. 

(7)  If  6%  stock  is  bought  at  109|,  what  per  cent  does  it 
pay  on  the  investment,  brokerage  J%  '? 

1  share  costs  -f  109|  +  $  ^  =  $  110. 

$  110  yields  an  income  of  $6. 

.'.  the  rate  per  cent  =  j^f^  or  5j\  %  of  the  investment. 

(8)  What  must  I  pay  for  6%  stock  to  realize  an  income  of 
8%  on  the  investment,  brokerage  |^%  ? 

8  %  of  the  cost  of  1  share  =  $  6. 
1  %  of  the  cost  of  1  share  =  $  |. 
100  %  of  the  cost  of  1  share  =  $  75  ; 
i.e.  including  brokerage,  the  cost  price  is  $  76. 
.•.  stock  is  quoted  at  74|. 

Exercise  196 

1.  What  will   25   shares  of   111.  Central   stock   cost  at   148, 
brokerage  |%  ? 

2.  What  is   realized  from  the  sale  of  208  shares  of  Chi.  & 
Alton  pfd.  R.  R.  stock  at  71|-,  brokerage  i%  ? 

2a 


354  ARITHMETIC 

3.  What  did  I  pay  for  39  shares  Union  Pacific,  Oct.  11,  1901, 
stock  selling  at  99^  and  brokerage  being  1%? 

4.  Find  what  I  received  from  the  sale  of  84  shares  of  Western 
Union  stock  at  92^,  brokerage  J%. 

5.  What  is  the  cost  of  ^20,000  U.S.  4's  at  112f,  brokerage 

6.  Find  the  cost  of  $  24,000  U.  S.  4's  at  116f,  brokerage  i%. 

7.  October  11,  1901,  96  shares  of  Pullman  stock  at  214|  were 
sold  on  the  New  York  stock  exchange,  brokerage  |%.  Find  the 
amount  received  by  the  owners  of  the  stock. 

8.  How  many  shares  of  Atchison  pfd.  stock  at  97J  can  I  buy 
for  ^3505.50,  brokerage  -i-%  ? 

9.  October  11,  1901,  Bal.  and  Ohio  stock  was  quoted  at  lOlJ. 
How  many  shares  were  bought  for  ^4257.75,  brokerage  J%  ? 

10.  A  stockholder  sold  D.  L.  and  W.  R.  R.  stock  at  157^, 
receiving  all  together  $3771.  How  many  shares  did  he  sell, 
brokerage  being  J  %  ? 

11.  How  many  shares  of  U.  S.  Steel  pfd.  stock  must  I  sell  at 
95,  brokerage  }%,  to  receive  $9677.25? 

12.  If  from  my  sales  of  Western  Union  stock  at  95J,  I  receive 
$6278.25,  how  much  stock  did  I  sell,  brokerage  being  l%? 

13.  How  many  shares  of  N.  Y.  Central  stock  at  158 J  can  be 
bought  for  $2855.25,  brokerage  i%? 

14.  A  broker  sells  24  shares  of  stock  on  a  commission  of  ^%. 
How  much  does  he  realize  ? 

15.  Plow  many  shares  of  stock  does  a  broker  sell  to  realize  a 
commission  of  $  16.25,  brokerage  ^%? 

16.  A  broker  realizes  $  12.50  from  the  sale  of  stock,  brokerage 
1%.  What  was  the  par  value  of  the  stock  sold  and  what  did  it 
sell  forat  70f? 

17.  A  broker  received  $  46.50  for  buying  stock  on  a  oommis- 
How  much  stock  did  he  buy  ? 


STOCKS   AND  BONDS  355 

18.  I  sold  through,  my  broker  40  shares  of  stock,  receiving  for 

it  f  4860,  brokerage  1%.     At  what  price  was  the  stock  quoted  ? 

19.  A  person  received  $6053.121  for  $  6500  stock  after  paying 
his  broker  -|-%.  Find  at  what  per  cent  discount  the  stock  was 
sold. 

20.  October  10,  1901,  $1654.25  was  paid  for  26  shares  of 
Pacific  Coast  stock,  brokerage  i%.  At  what  was  Pacific  Coast 
stock  quoted,  Oct.  10  ? 

21.  What  annual  income  will  be  obtained  from  $  6071,  invested 
in  U.  S.  4's  coup,  of  1925  at  116f,  brokerage  i%  ? 

22.  A  person  paid  $8578.50  for  U.  S.  4's  at  112f,  brokerage 
i%.     What  was  his  income  from  the  bonds  ? 

23.  If  I  invest  $8583.75  in  stock  at  95J,  paying  5%  dividend, 
what  will  be  my  income,  brokerage  ^%  ? 

24.  What  income  will  be  realized  from  $  9229.50  invested  in 
stock  at  109f,  brokerage  ^%,  paying  a  dividend  of  5^%  ? 

25.  What  amount  of  money  must  be  invested  in  8%  stock  at 
158 J,  brokerage  i%,  to  realize  an  income  of  $  1096  ? 

26.  What  sum  must  I  invest  in  4^%  stock  at  99|,  to  produce 
an  annual  income  of  $  1638,  brokerage  ^%? 

27.  How  much  must  I  invest  in  U.  S.  5's  at  112J  to  realize 
an  annual  income  of  $450,  brokerage  -g-%  ? 

28.  If  street  railway  stock  bought  at  232  yields  a  half-yearly 
dividend  of  6i%,  how  much  must  I  invest  to  obtain  a  semiannual 
income  of  $325,  brokerage  i%  ? 

29.  If  I  buy  stock  through  a  broker  who  charges  |^%,  how 
much  must  I  invest  in  stock  at  153,  paying  9%  dividends,  to 
secure  an  income  of  $  1350  ? 

30.  If  4|%  stock  is  bought  at  74|-,  brokerage  i%,  what  per 
cent  does  it  pay  on  the  investment  ? 

31.  If  8%  stock  is  bought  at  159|,  what  per  cent  does  it  pay 
on  the  investment,  brokerage  ^%? 


356  ARITHMETIC 

32.  What  must  I  pay  for  4%  stock  to  pay  5%  on  the  invest- 
ment, brokerage  ^%? 

33.  What  rate  of  interest  do  I  realize  on  an  investment  in  6% 
stock  at  107 J,  brokerage  |%  ? 

34.  What  must  I  pay  for  5%  stock  to  yield  an  income  of  6% 
on  my  investment  ? 

35.  A  person  receives  $  600  from  an  8%  bank  dividend.  How 
much  stock  does  he  own  ? 

36.  A  person  having  $  5000  bank  stock  sells  out  when  it  is  at 
40%  premium.    What  amount  of  money  does  he  receive,  brokerage 

being  i%? 

37.  Bought  through  a  broker  1600  shares  ($  100)  R.  R.  stock  at 
69 J,  brokerage  |^%.     What  was  the  cost  of  the  stock  ? 

38.  A  speculator  bought  36,500  shares  {$  100)  R.  R.  stock  at 
39|,  and  sold  them  at  40f.  What  was  his  gain,  brokerage  |^%,  on 
both  transactions  ? 

39.  A  bank  declared  a  dividend  of  3J%.  How  much  should  a 
stockholder  owning  120  shares  ($  50)  receive  ? 

40.  One  company  guarantees  to  pay  6%  on  shares  of  $100 
each;  another  guarantees  at  the  rate  of  5|%  on  shares  of  $30 
each ;  the  price  of  the  former  is  $  124.50,  and  of  the  latter  $  34. 
Find  the  rates  of  interest  which  they  return  to  the  purchaser. 

41.  A  broker  receives  $42,100  to  invest  in  U.  S.  5-20  bonds, 
after  reserving  J%  on  the  par  value  of  the  amount  purchased. 
What  was  his  commission,  the  bonds  being  at  a  premium  of  5%  ? 

42.  A  man  bought  through  a  broker  1900  shares  ($100)  R.  R. 
stock  at  54|  and  sold  them  at  55f .  What  was  his  net  profit  on 
the  transaction,  brokerage  each  way  \%  ? 

43.  An  insurance  company  declared  a  dividend  of  9%.  What 
rate  is  that  on  the  market  value  of  the  shares  which  are  at  185  ? 

44.  Compare  the  rates  on  the  cash  values  of  6%  on  stock  at 
216  and  3|%  on  stock  at  125. 


COMPOUND   PROPORTION  357 

45.  Sold  37  shares  ($  25)  B.  and  L.  Association  stock,  receiving 
therefor  $  1019.81.     At  what  rate  was  the  stock  sold  ? 

46.  Bought  through  a  broker  750  shares  ($  50)  in  the  Farmers' 
Loan  and  Savings  Society,  paying  therefor  $43,968.75.  At  what 
quotation  were  they  bought,  brokerage  |^%  ? 

47.  Bought  stock  at  197|  and  sold  it  at  194-|,  having  mean- 
while received  a  dividend  of  69^  on  it.  My  net  gain  on  the 
transaction  after  paying  1%  brokerage  each  way  is  $336.  How 
many  shares  ($  40)  did  I  buy  ? 

48.  How  many  railway  shares  ($100)  at  40%  discount  must 
be  sold  in  order  that  the  proceeds  invested  in  bank  stock,  which 
is  4%  below  par,  and  pays  a  dividend  of  7%,  may  yield  an  income 
of  $  1680,  brokerage  included  ? 

49.  Explain  the  terms :  Stocks,  Shares,  Dividends.  When  is 
stock  at  par  ?     At  a  premium  ?     At  a  discount  ? 

50.  When  the  3|-  per  cents  are  at  98,  what  must  be  the  price  of 
another  stock  yielding  ^%,  so  that  the  latter  may  be  as  profit- 
able as  the  former,  brokerage  included  ? 

COMPOUND  PROPORTION 

276.  (1)  If  20  men  can  dig  60  yd.  of  earth  in  4  da.,  how 
many  yards  can  30  men  dig  in  9  da.  ? 

Men  Yd.  Da. 

20  60  4 

30  x  9 

Multiply  60  yd.  by  f^.     •.•  .SO  men  can  dig  |§  as  many  yards  as  20  men. 
Multiply  the  result  by  f ,     •.•  in  9  da.  30  men  can  dig  |  as  much  as  in  4  da. 

.-.  X  yd.  =  -s^fi  yd.  X  f ^  X  f  =  202^  yd. 

(2)  If  120  bu.  of  oats  last  14  horses  5Q  da.,  in  how  many 
days  will  6  horses  consume  90  bn.  ? 


Bu. 

Horses 

Da. 

120 

14 

66 

90 

6 

X 

358  ARITHMETIC 

Multiply  56  da.  by  ~^^.     •.•  90  bu.  will  last  ^^^  as  many  days  as  120  bu. 
Multiply  the  result  by  ^.     •/  90  bu.  will  last  0  horses  ^  ^  long  as  14 
horses. 

.-.  X  da.  =  5/  da.  X  T»3^  X  V  =  ^8  da. 

277.  In  each  of  the  above  two  solutions,  in  order  that  the 
ratio  may  easily  be  seen,  the  items  in  the  question  have  been 
written  in  horizontal  lines.  In  number  (1)  we  are  required 
to  find  the  number  of  yards,  and  the  problem  is  to  determine 
the  ratio  resulting  from  each  comparison,  and  how  it  affects 
the  number  of  yards. 

In  number  (2)  we  are  required  to  find  the  number  of  days, 
and  the  problem  is  to  determine  the  ratios,  and  how  they 
affect  the  number  of  days. 

278.  To  prove  the  answer  correct,  substitute  the  answer 
in  place  of  x  in  the  horizontal  line  and  omit  one  of  the  quan- 
tities, frame  the  question,  and  then  solve. 


5u. 

Horses 

Da. 

.20 

14 

66 

90 

X 

98 

If  120  bu.  of  oats  last  14  horses  for  bQ  da.,  how  many 
horses  will  90  bu.  last  98  da.  ? 

On  solving,  x  will  be  found  equal  to  6,  which  proves  the  former  solution 
correct.  How  many  questions  can  be  made  from  the  numbers  in  the  two 
lines,  including  the  original  one  ? 

Solve  the  following  questions.  State  one  or  more  questions 
in  proof  for  each  problem,  and  prove  your  answers  correct. 

Exercise  197 

1.  If  7  horses  are  kept  20  da.  for  $  14,  how  many  will  be  kept 
7  da.  for  ^28? 

2.  If  3  men  earn  $75  in  20  da.,  how  many  men  will  earn 
$  78.75  in  9  da.  at  the  same  rate  ? 


COMPOUND   PROPORTION  359 

3.  If  16  horses  eat  96  bii.  of  corn  in  42  da.,  in  how  many  days 
will  7  horses  eat  66  bu.  ? 

4.  If  16  horses  can  plough  1280  A.  in  8  da.,  how  many  acres 
will  12  horses  plough  in  5  da.  ? 

5.  If  20  men  can  perform  a  piece  of  work  in  12  da.,  find  the 
number  of  men  who  could  perform  another  piece  of  work  3  times 
as  great  in  4-  of  the  time. 

6.  If  252  men  can  dig  a  trench  210  yd.  long,  3  wide,  and  2 
deep  in  5  da.  of  11  hr.  each,  in  how  many  days  of  9  hr.  each  will 
22  men  dig  a  trench  of  420  yd.  long,  5  wide,  and  3  deep  ? 

7.  If  10  men  can  reap  a  field  of  7|-  A.  in  3  da.  of  12  hr.  each, 
how  long  will  it  take  8  men  to  reap  9  A.,  working  16  hr.  a  day  ? 

8.  If  25  men  can  do  a  piece  of  work  in  24  da.,  working  8  hr. 
a  day,  how  many  hours  a  day  would  30  men  have  to  work  in  order 
to  do  the  same  piece  of  work  in  16  da.  ? 

9.  A  town  which  is  defended  by  1200  men,  with  provisions 
enough  to  sustain  them  42  da.,  supposing  each  man  to  receive 
18  oz.  a  day,  obtains  an  increase  of  200  men  to  its  garrison. 
What  must  now  be  the  allowance  to  each  man,  in  order  that  the 
provisions  may  serve  the  whole  garrison  for  54  da.  ? 

10.  If  560  flagstones,  each  l-i-  ft.  square,  will  pave  a  court- 
yard, how  many  will  be  required  for  a  yard  twice  the  size,  each 
flagstone  being  14  in.  by  9  in.  ? 

11.  If  20  men  in  3  wk.  earn  $  900,  in  what  time  will  12  men 
earn  $  1500  ? 

12.  If  -f-f  of  a  meadow  be  mown  by  12  men  in  6  da.,  find  in 
what  time  the  remainder  could  be  mown  by  10  men. 

13.  If  36  men,  working  16  da.,  can  dig  a  trench  72  yd.  long, 
18  yd.  wide,  and  12  yd.  deep,  how  many  mea  can  dig  a  trench 
64  yd.  long,  27  yd.  wide,  and  18  yd.  deep  in  24  da.  ? 

14.  If  25  men  build  a  wall  15  ft.  high,  2  ft.  thick,  and  50  ft. 
long,  in  12  da.  of  9  hr.  each,  how  many  hours  per  day  must  40 
men  work  to  build  a  wall  60  ft.  long,  3  ft.  thick,  and  20  ft.  high 
in  27  da.  ? 


360  ARITHMETIC 

15.  A  miller  has  a  bin  8  ft.  long,  41  ft.  wide,  and  2^  ft.  deep, 
holding  68  bu.  How  deep  must  he  make  another  bin  which  is  to 
be  18  ft.  long  and  3J  ft.  wide,  so  that  its  capacity  may  be  408  bu.  ? 

16.  What  is  the  weight  of  a  block  of  stone  12  ft.  6  in.  long, 
6  ft.  6  in.  broad,  and  8  ft.  3  in.  deep,  when  a  block  of  the  same 
stone  5  ft.  long,  3  ft.  9  in.  broad,  and  2  ft.  6  in.  deep,  weighs 
7500  lb.  ? 

COMPOUND  PARTNERSHIP 

279.  In  Compound  Partnership  the  time  is  taken  into 
account  as  well  as  the  capital  in  determining  the  gain  or 
loss  of  each  partner. 

280.  (1)  A,  B,  and  C  enter  into  partnership.  A  puts  in 
1700  for  12  mo.,  B  1 500  for  9  mo.,  and  C  $600  for  8  mo. 
Divide  a  profit  of  1 2065  equitably  among  them. 

The  gain  on  $  700  for  12  mo.  =  the  gain  on  $  8400  for  1  mo. 

The  gain  on  .$  500  for    9  mo.  =  the  gain  on  $  4500  for  1  mo. 

The  gain  on  $  600  for    8  mo.  =  the  gain  on  $  4800  for  1  mo. 
.'.  the  proportional  parts  representing  the  gains  are  84,  45,  and  48,  or  28, 
15,  and  16.  28  +  15  +  16  =  59. 

.-.  the  respective  gains  are  |f ,  ^f ,  and  f  |  of  .$  2065  =  $  980,  $  525,  and  §  560. 

(2)  A  commenced  business  with  $  4000  stock ;  3  mo.  after, 
he  took  in  B  with  a  capital  of  $  2000 ;  and  4  mo.  after  B 
became  a  partner,  he  took  in  C  with  a  capital  of  f  600 ;  at 
the  end  of  the  year  the  firm  had  gained  $3450.  Find  the 
share  of  each. 

A's  capital  =  $4000  for  12  mo.  =  $48,000  for  1  mo. 
B's  capital  =  $2000  for    9  mo.  =  $  18,000  for  1  mo. 
C's  capital  =  $  600    for    5  mo.  =  $  3000  for  1  mo. 
.'.  the  respective  gains  are  proportional  to  48,  18,  and  3 ;  i.e.  16,  6,  and  1. 

16  +  6  +  1  =  23. 
.-.  the  respective  shares  are  Jf,  j\,  and  ^  of  $3450  =  $2400,  $900,  and 
$  150. 


COMPOUND  PARTNERSHIP  361 

Exercise  198 

1.  D  and  E  enter  into  partnership ;  D  puts  in  $  480  for  3  mo., 
and  E  $  900  for  4  mo.  They  gain  $  840.  What  is  each  man's 
share  in  the  gain  ? 

2.  A,  B,  and  C  entered  into  partnership;  A  put  in  ^1200  for 

8  mo.,  B  ^800  for  10  mo.,  and  C  $400  for  12  mo.     They  gained 
$  3920.     What  was  each  man's  share  of  the  gain  ? 

3.  A,  B,  and  C  are  partners;  A  puts  in  $5000  for  6  mo., 
B  $6000  for  8  mo.,  and  C  $900  for  11  mo.  The  profit  is 
$5575.50.     What  is  the  share  of  each? 

4.  Three  graziers  hire  a  pasture  for  their  common  use,  for 
which  they  pay  $  318.  One  puts  in  20  oxen  for  6  mo.,  another 
24  oxen  for  8  mo.,  and  the  third  28  oxen  for  4  mo.  How  much 
of  the  rent  should  each  pay  ? 

5.  A  and  B  enter  into  partnership ;  A  contributes  $  15,000  for 

9  mo.,  and  B  $12,000  for  6  mo.     They  gain  $5750.     Find  each 
man's  share  of  the  gain. 

6.  A,  B,  and  C  rent  a  field  for  $  56.50 ;  A  puts  in  70  cattle 
for  6  mo.,  B  40  for  9  mo.,  and  C  50  for  7  mo.  What  ought  each 
to  pay  ? 

7.  Three  merchants  enter  into  partnership ;  the  first  invests 
$1855  for  7  mo.,  the  second  invests  $887.50  for  10  mo.,  and  the 
third  invests  $  770  for  11  mo.,  and  they  gain  $434.  What  should 
be  each  partner's  share  of  the  gain  ? 

8.  L,  M,  and  N  entered  into  partnership  and  invested  respec- 
tively $19,200,  $22,500,  and  $28,300.  At  the  end  of  5  mo.  L 
invested  $  3800  additional,  M  $  2500,  and  N  $  3700.  At  the  end 
of  a  year  the  net  gain  of  the  firm  was  found  to  be  $  7850.  What 
was  each  partner's  share  of  this  ? 

9.  A  and  B  enter  into  partnership;  A  puts  in  $400  at  first, 
and  $500  at  the  end  of  2  mo. ;  B  puts  in  $300  at  first,  and 
$600  at  the  end  of  3  mo.  The  profit  at  the  end  of  the  year 
is  $  470.     How  should  this  be  divided  ? 


362  ARITHMETIC 

10.  A  and  B  engage  in  trade:  A  invests  $6000,  and  at  the 
end  of  5  mo.  withdraws  $  2000 ;  B  puts  into  the  business  $  4000, 
and  at  the  end  of  7  mo.  $6000  more.  Divide  a  gain  of  $6800  at 
the  end  of  the  year. 

11.  A,  B,  and  C  form  a  partnership  with  a  joint  stock  of 
$15,600;  A's  stock  continues  in  trade  6  mo.,  B's  8  mo.,  and 
C's  12  mo.  A^s  gain  is  $1200,  B's  $2400,  and  C's  $1800.  What 
stock  did  each  put  in  ? 

12.  Two  men  complete  in  a  fortnight  a  piece  of  work  for  which 
they  are  paid  $  46.75 ;  one  of  them  works  alternately  9  hr.  and 
8  hr.  a  day,  the  other  works  8^  hr.  for  5  da.  in  the  week,  and 
does  nothing  on  the  remaining  day.  What  sum  should  each 
receive  ? 

13.  A  and  B  are  partners ;  A's  capital  is  to  B's  as  4  to  9.  At 
the  end  of  4  mo.  A  withdraws  ^  of  his  capital,  and  B  |^  of  his. 
At  the  end  of  the  year  their  whole  gain  is  $  4600.  How  much 
belongs  to  each  ? 

CUBE   ROOT 

281.  The  product  of  3  x  3  x  3  is  27 ;  of  5  x  5  x  5  is  125. 
The  cubes  whose  sides  measure  3  and  5  units  of  length  con- 
tain 27  and  125  units  of  volume.  We  say  that  27  is  the  cube 
of  3  and  that  125  is  the  cube  of  5 ;  that  3  is  the  cube  root 
of  27,  and  that  5  is  the  cube  root  of  125.  The  cube  of  6  is 
written  5^  and  the  cube  root  of  5  is  indicated  thus :  V5. 

5^  is  also  called  the  third  power  of  5. 

282.  The  cubes  of 

1,    2,     3,      4,       5,        6,        7,        8,        9,        10, 
are   1,    8,    27,    64,    125,    216,    343,    512,    729,    1000. 

283.  The  cube  roots  of 

1,    8,    27,    64,    125,    216,    343,    512,    720,    1000, 
are   1,    2,     3,      4,       5,        6,        7,        8,        9,        10. 


CUBE  ROOT  363 

284.  These  two  paragraphs  should  be  mastered  by  the 
pupils  as  the  corresponding  paragraphs  in  square  root. 

285.  The  product  4  x  4  x  4  is  written  4^. 

The  product  4.6  x  4.6  x  4.6  is  written'  (4.6)3. 
The  product  f  x  J  X  |  is  written  (|)3. 
The  cube  root  of  4  is  written  V 4. 
■  The  cube  root  of  .4  is  written  V.4. 
The  cube  root  of  |  is  written  "v|. 

Exercise  199 
Write  the  following  products  as  powers : 

1.2x2x2.  7.  4  X  I  X  |. 

2.  3  X  3.  8.  I  X  I  X  |. 

3.  5  X  5  X  5.  9.  2.3  X  2.3  x  2.3. 

4.  5x5x5x5.  10.  3.12  x  3.12  x  3.12. 

5.  5  X  5  X  5  X  5  X  5.  ii.  .12  x  .12  x  .12. 

6.  5x5x5x5x5x5.  12.  .1  x  .1  x  .1. 

13.    .02  X  .02  X  .02. 

Write  the  following  powers  as  products  and  find  their  values : 
14.    43.  15.    12^.  16.    2.5\  17.    (1)^  18.    {^y. 

19.    .02^  20.    .1^ 

Prove  the  following  statements  by  multiplication : 

21.    ^/2l6  =  6.  26.    ^:064  =  .4. 


22. 

a/15625  =  25. 

27. 
28. 
29. 
30. 

^S  =  |. 

23. 

V'15.625  =  2.5. 

^iif  =  f 

24. 
25. 

a/1.728  =  1.2. 
^.008  =  .2. 

364  ARITHMETIC 

Exercise  200 

1.  Find  the  length  of  one  edge  of  a  cube  containing  512  cu. 
in.  Find  the  length  of  all  its  edges.  Find  the  area  of  one  of  its 
faces.     Of  all  its  faces. 

2.  Find  the  area  of  one  face  of  a  cube  containing  729  cu.  in. 

3.  Find  the  number  of  units  of  length  in  a  cube  containing 
343  units  of  volume.  Find  the  number  of  units  of  area  in  one 
face. 

4.  Find  the  edge  of  a  cube  one  of  whose  faces  contains  144 
sq.  in.     Find  its  volume. 

5.  Find  the  volume  of  a  cube  one  of  whose  faces  contains 
225  sq.  in. 

6.  What  is  the  edge  of  a  cube  whose  volume  is  8  units  of  vol- 
ume ?     27  units  of  volume  ? 

7.  The  ratio  of  the  volumes  of  two  cubes  is  -^.  What  is  the 
ratio  of  their  edges  ? 

8.  •  The  ratio  of  the  volumes  of  two  cubes  is  64 :  125.  What  is 
the  ratio  of  their  edges  ? 

9.  The  ratio  of  the  edges  of  two  cubes  is  ^.  What  is  the 
ratio  of  their  volumes  ? 

10.   The  edges  of  two  cubes  are  as  7:9.     What  is  the  ratio  of 
their  volumes  ? 

Exercise  201 

1.  Find  the  cubes  of  14,  25,  36,  54,  75,  and  99. 

2.  From  the  results  in  §  282,  state  how  many  digits  there  are 
in  the  cube  of  a  number  of  1  digit. 

3.  From  the  results  in  example  1,  state  how  many  digits  there 
are  in  the  cube  of  a  number  of  2  digits. 

4.  In  long  division  how  many  figures  form  a  group?  In 
square  root  ?     In  cube  root  ? 

5.  How  many  digits  are  there  in  the  cube  root  of  612  ?  64  ? 
8  ?     What  are  they  ? 


CUBE  ROOT  365 

6.  Divide 389,017  into  groups  of  figures;  29,791;  3375.    How 
many  figures  are  there  in  the  cube  root  of  each  of  these  numbers  ? 

7.  Cube  73,  31,  and  15. 

8.  Judging  from  the  results  given  in  §  283,  state  the  number 
of  digits  in  the  cube  root  of  a  number  containing  1,  2,  or  3  digits. 

9.  Write  the  cubes  of  10,  20,  30,  40,  50,  60,  70,  80,  and  90. 

10.  State  how  many  digits  there  are  in  the  cube  root  of  a  num- 
ber containing  4,  5,  or  6  digits. 

11.  What  is  the  first  digit  in  the  cube  root  of  2744  ?     39,304  ? 

286.    To  find  the  cube  root  of  a  number,  we  shall  first  see 
how  the  cube  of  a  number  is  found. 

Since  54  =  50  +  4,  we  can  cube  54  thus  : 

50  +  4 
60  +  4 


(4  X  50)  +  42 
502  +  (4  X  50) 
502  _|.  2(4  X  50)  +  42 

50  +  4 

(4  X  502)  +  2(42  X  50)  +  43 
50^  +  2(4x502)+    (42x50) 
503  +  3(4  X  502)  4.  3(42  X  50)  +  4^ 

Since  4  divides  each  of  the  last  three  terms,  we  can  put  this  result 
='503  +  4{8  x  502  +  3  X  4  X  50  +  42^. 

We  now  wish  to  recover  from  such  a  number  as  157,464  its  cube  root. 
Plainly  the  tens'  digit  of  the  root  is  5,  i.  e.  the  first  part  of  the  root  is  50. 


50 

3  X  502  =  7500 

3  X  4  X  50  =    600 

42=      16 

8116 


157'464|50  +  4 
125000 


32464 


32464 


To  find  the  second  term,  note  that  in  the  expression 
4{3  X  502  +  3  X  4  X  50  +  42] 


366 


ARITHMETIC 


the  number  4  is  the  second  digit  in  the  number  54  that  was  cubed  ;  hence  in 
the  work  of  taking  the  cube  root,  the  other  factor  {3  x  bO^  +  3  x  4  x  50  -f  4^} 
will  be  the  real  divisor  and  3  x  50^  the  trial  divisor. 

Therefore,  squaring  50  and  multiplying  by  3,  we  have  7500.  Dividing  7500 
into  32,464,  we  find  the  quotient  to  be  4  ;  completing  the  divisor  by  adding 
3  X  4  X  50  or  600,  and  i'^  or  16,  we  find  the  divisor  to  be  7500  +  600  +  16 
or  8116.  Multiplying  this  by  4,  we  have  32,464.  Hence  we  conclude  that 
the  cube  root  of  157,464  is  54.     To  prove  the  result  correct,  cube  54. 


287.  The  work  of  extracting  the  cube  root  may  be  shortened 

thus : 

157'464[54 


300  X  52  =  7500 

30  X  5  X  4  =  600 

42  =  __16 

8116 


125 


32464 


32464 


Find  the  cube  root  of  926,859,375. 


300  X  92  =  24300 
30  X  9  X  7  =  1890 

72  = 49 

26239 

300  X  972  =  2822700 

30  X  97  X  5  =   14550 

52  = 25 

2837275 


926^859^3751975 
729 


197859 


183673 


14186375 


14186375 


288.  The  cube  of  3.19  is  equal  to  32.461759.  From  this 
it  is  evident  that  corresponding  to  the  two  figures  in  the 
decimal  part  of  the  number,  viz.  19,  we  have  two  groups  of 
three  figures,  viz.  461  and  759,  in  the  decimal  part  of  the 
cube.  Hence  in  pointing  off,  begin  at  the  decimal  point  and 
mark  the  number  off  into  periods  of  tliree  figures  each  to  the 
right  of  the  decimal  and  then  again  to  the  left.  Mark  off 
into  periods  the  number  95.443993. 


CUBE  ROOT 


367 


(1)  Find  the  cube  root  of  95.443993. 


300  X  42  =  4800 

30  X  4  X  5  =  600 

52=   25 

5425 

300  X  452  =  607500 

30  X  45  X  7  =   9450 

72=    49 


95.443'993  |  4.57 
64 


31443 


27125 


4318993 


4818993 


616999 

(2)  Extract  the  cube  root  of  16. 

16  I  2.519 


300  X  22  ^  1200 

30  X  2  X  5  =  300 

52=  _25 

1525 

300  X  252  =  187500 

30  X  25  X  1  =   750 

12=     1 


188251 

300  X  2512  =  18900800 

30  X  251  X  9  =  667700 

92=      81 


19508081 


8000 


7625 


375000 


188251 


186749000 


176112729 


(3)  To  extract  the  cube  root  of  such  a  number  as  843.7295, 
add  ciphers  thus,  843.729'500,  and  extract  the  cube  root. 

(4)  Extract  the  cube  root  of  3^%. 


>'343 


^64 


Sy343 

(5)  Extract  the  cube  root  of  -^^^. 


«'16 
A/343 


^16       2.519 


343      ^343 


=  .359. 


368  ARITHMETIC 

(6)  Extract  the  cube  root  of  J|. 

if  =  .64  or  .640. 

\/.640  =  .861. 

•••  Vii  =  .861. 

Which  of  the  three  denominators  is  not  a  perfect  cube  ?  When  should 
a  fraction  be  reduced  to  a  decimal  before  extracting  its  cube  root  ?     Why  ? 

Exercise  202 
Find  the  cube  root  of : 
1.   29,791.  2.    54,872.  3.    110,592. 

4.    804,357.  5.    941,192. 

6.  Compare  the  processes  of  long  division,  of  extracting  the 
square  root,  and  the  cube  root  of  a  number.  Note  in  what  respect 
they  are  similar  and  in  what  respect  they  are  different. 

7.  2,406,104.  13.    .001906624. 

8.  69,426,531.  14.   3,  .3,  .03,  .003,  .0003. 

9.  8,365,427.  15.    ^,  |||,  3^. 

10.  389.017.  16,    11^,  «f. 

11.  32.461759.  17.   |,  ^jV^,  f 

12.  .000912673.  18.   3|,  405t\V»  7f 

19.  A  cubical  block  of  stone  contains  50,653  cu.  ft.  What  is 
the  area  of  its  side? 

20.  A  cube  contains  56  cu.  ft.  568  cu.  in.     Find  its  edge. 

21.  One  gallon  contains  231  cu.  in.  Find  the  edge  of  a  cube 
equal  to  it. 

22.  Find  the  length  of  the  inside  edge  of  a  cubical  vessel  which 
will  just  hold  10  gal. 

23.  Three  cubes  of  lead,  measuring  respectively  ^,  f,  and  -J  of 
an  inch  on  the  edge,  were  melted  together  and  cast  into  a  single 
cube.  Find  the  length  of  the  edge  of  the  cube  thus  formed, 
neglecting  loss  of  lead  in  melting  and  casting. 


CUBE  ROOT  369 

24.  Four  cubes  of  lead,  measuring  respectively  6,  7,  8,  and  9  in. 
on  the  edge,  were  melted  together  and  cast  into  a  single  cube. 
Find  the  length  of  the  edge  of  the  cube  thus  formed,  if  4%  of  the 
lead  was  lost  in  melting  and  casting. 

25.  Find  the  volume  of  a  cube,  the  area  of  whose  surface  is 
100.86  sq.  in. 

26.  A  cube  measures  5  in.  on  the  edge.  A  second  cube  is  3 
times  the  volume  of  the  first.  By  how  much  does  the  length  of 
an  edge  of  the  second  cube  exceed  that  of  an  edge  of  the  first 
cube? 

27.  By  raising  the  temperature  of  a  cube  of  iron  the  length 
of  each  of  its  edges  was  increased  by  5%.  Find  correct  to  four 
decimals  the  ratio  of  increase  in  the  volume  of  the  cube. 

28.  Each  edge  of  a  cube  is  diminished  by  -^q  of  its  length. 
By  what  fraction  of  itself  is  the  volume  diminished  ?  By  what 
fraction  of  itself  is  the  area  of  the  surface  diminished  ? 


THE  PUBLIC  SCHOOL  ARITHMETIC 

Based  on  McLellan  and  Dewey^s  "Psychology  of  Number'' 


By  J.  A.  McLELLAN 

President  of  the  Ontario  Normal  College 

AND 

A.  F.  AMES,  A.B. 

Superintendent  of  Schools,  Riverside,  III. 

Cloth.     i2nio.    60  cents,  net 


"  Naturally  I  am  pleased  with  the  extent  to  which  the  book  bases  the  treatment  of 
fundamental  operations  of  fractions  and  ratio  upon  the  idea  of  measure  and  of  num- 
bers as  units  of  measurement.  I  am  particularly  struck  with  the  fact  that  the  pupil's 
attention  is  definitely  called  to  some  special  quantity  or  whole  which  furnishes  the 
object  of  attention,  and  within  which,  so  to  speak,  the  numerical  processes  take  place; 
also  with  the  clearness  and  conciseness  of  the  method  of  treatment ;  the  logical  order 
of  the  selection  of  topics ;  and  the  exclusion  of  useless  and  irrelevant  matter.  The 
simplification  of  treatment  due  to  sticking  close  \o  fundamental  principles  must  recom- 
mend the  book  to  teachers  and  pupils  who  have  been  bewildered  by  the  great  number 
of  topics  treated  in  the  ordinary  arithmetic  —  topics  which  do  not  differ  at  all  in  their 
logical  or  arithmetical  basis,  but  are  simply  different  practical  expressions  of  the  same 
principle.     I  wish  the  book  the  success  it  deserves." 

—  Professor  Dewey,  University  of  Chicago. 

"The  idea  of  number  as  measurement  is  treated  with  great  thoroughness  both 
in  the  abundance  of  explanation  and  illustration,  and  in  the  number  of  exercises. 
Every  lesson  teaches  with  definiteness  the  points  it  is  intended  to  make  clear,  and  in 
each  chapter  there  is  a  reference  to  a  particular  quantity,  so  that  the  child's  ideas  are 
made  definite  from  the  beginning.  Everything  in  the  lessons  bears  on  the  topic,  and 
there  is  no  unnecessary  matter  taken  from  other  books  to  fill  it  out.  Many  teachers 
who  are  tired  of  the  mechanical,  uninteresting  way  arithmetic  has  been  taught  will 
find  here  a  natural  treatment  of  the  subject  that  will  prove  for  themselves  a  stimulus 
in  their  work,  and  be,  for  their  pupils,  a  means  of  making  the  exercises  pleasant  and 
easy."  —  ALFRED  M.  BURCHILL,  Principal  Training  Institute,  Burke's  Falls,  Can. 

"Since  I  obtained  a  copy  of  the  'Psychology  of  Number'  I  have  been  a  close 
student  of  its  statements,  and  no  book  on  arithmetic  has  ever  done  me  half  as  much 
good  for  my  work.  Opponents  may  criticise  it,  but  the  fact  remains  that  it  opens  up 
to  the  student  lines  of  thought  that  are  fruitful  indeed.  Therefore  I  secured  a  copy  of 
the  '  Public  School  Arithmetic '  since  it  is  based  upon  the  former.  It  is  more  than 
helpful;  it  is  broadening,  it  is  stimulating.  It  brings  to  light  rational  method,  and 
grasps  number  as  measurement.  I  am  especially  pleased  with  the  typical  solutions." 
—  Charles  E.  Hutton,  Principal  Los  Angeles  Normal  School. 


THE    MACMILLAN    COMPANY 

NEW  YORK         CHICAGO         BOSTON  SAN  FRANCISCO  ATLANTA 


THE  PRIMARY  PUBLIC  SCHOOL 
ARITHMETIC 

By  J.  A.  McLELLAN 

President  of  the  Ontario  Normal  College 

AND 

A.  F.  AMES,  A.B. 
Superintendent  of  Schools,  Riverside,  III. 

Cloth.     I2I110.    35  cents,  net 
Also  an  Edition  for  Teachers.    Cloth.     i2nio.    50  cents,  net 


"  I  am  delighted  with  the  '  Primary  Arithmetic'  The  authors  have  succeeded  in 
putting  into  practical  shape,  for  junior  pupils,  the  principles  laid  down  in  the  '  Psy- 
chology of  Number.'  In  the  teachers'  edition  the  sugg;estions  to  teachers  and  the  sug- 
gestive lessons  are  invaluable,  and  the  arithmetic  proper  has  the  exercises  so  well 
arranged,  and  put  in  such  practical  form,  that  pupils  will  at  once  become  deeply  inter- 
ested in  this  subject.  Very  rapid  advancement  will  consequently  be  made  by  them  in 
the  fundamental  principles  and  processes."  —  R.  A.  THOMPSON,  Hamilton,  Ont. 

"This  book  adds  another  to  the  list  of  valuable  school  texts  that  during  the  last 
twenty  years  have  come  from  the  pen  of  Dr.  McLellan.  For  the  first  time  in  any  arith- 
metic, the  teacher  is  shown  how  to  give  the  first  lessons  in  number  in  harmony  with 
the  psychical  nature  of  number  and  the  actual  development  of  number  concepts  in  the 
child's  mind.  The  book  is  founded  on  the  idea  of  measurement,  to  be  done  by  the 
child  himself,  so  that  the  symbol  of  number  comes  to  the  child  as  a  reward  for  work 
done  by  himself.  The  lessons  are  so  graded  that  each  seems  to  be  an  application  of 
the  preceding.  In  the  presentation  of  '  matter,'  regard  is  had,  in  the  method  adopted, 
to  the  normal  development  of  the  child's  mind.  By  using  the  idea  of  measurement  and 
comparison  from  the  first,  the  pupil  is  unconsciously  gaining  ideas  of  fractions  and 
percentage,  so  that,  when  he  comes  to  these  rules  he  begins  to  put  into  conscious  prac- 
tice ideas  unconsciously  known  before.  This  book,  in  the  hands  of  teachers  of  primary 
arithmetic  in  our  public  schools,  would  be  of  inestimable  value  as  a  means  of  suggest- 
ing methods  of  teaching  arithmetic  to  junior  pupils." 

—  Free  Press  (London,  Ontario,  Can.). 

"  More  than  any  other  primary  arithmetic  does  it  combine  the  scientific  foundation 
with  the  great  social  interest  that  is  such  an  aid  to  the  teacher  in  developing  this  sub- 
ject. ...  In  this  series  of  books  Dr.  McLellan  has  in  the  words  of  Horace, '  reared 
a  monument  more  enduring  than  brass,'  and  of  all  his  works  the  '  Primary  Arithmetic," 
which  lightens  the  burden  of  the  primary  school  teachers  and  makes  easier  and  more 
pleasant  the  pathway  of  the  little  child,  deserves  special  prominence." 

—  The  Globe  (Toronto,  Can.). 


THE    MACMILLAN    COMPANY 

NEW  YORK         CHICAGO         BOSTON  SAN  FRANCISCO  ATLANTA 


MENTAL  ARITHMETIC 

By  J.  A.  McLELLAN 

President  of  the  Ontario  Normal  College 


A.  F.  AMES,  A.B. 

Superintendent  of  Schools,  Riverside,  III, 

Cloth.    1 2 mo.    25  cents,  net 


The  present  authors'  "  Public  School  Arithmetic  "  and  the  "  Primary  Public  School 
Arithmetic  "  differ  from  all  other  text-books,  in  being  based  on  the  psychology  of  num- 
ber. The  "  Mental  Arithmetic,"  based  on  the  same  principles,  completes  the  series,  and 
completes  the  method.  As  compared  with  written  arithmetic  alone,  mental  arithmetic 
systematically  taught  will  produce  at  least  twice  the  knowledge  and  twice  the  power  in  a 
given  time.    The  distinguishing  features  of  the  book  are :  — 

1.  It  is  not  a  book  of  puzzles  for  ingenious  analysis,  but  a  book  of  principles  for 
easy  mastery  of  rational  method. 

2.  Like  the  two  books  for  written  arithmetic,  the  mental  arithmetic  is  based  on  the 
idea  of  number  as  measurement,  and  this  again  as  arising  from  human  needs.  There- 
fore, all  the  processes  have  meaning  for  the  pupil.  They  are  connected  with  his  own 
experience.  In  fact,  every  step,  from  first  to  last,  is  simply  a  making  over  his  own  ex- 
perience. In  the  first  lesson,  and  in  all  lessons,  the  child  is  always  learning  with  what 
he  has  learned. 

3.  This  means  that  there  is  continuity.  The  very  first  question  looks  toward  the 
very  last.  Every  "  new  rule  "  deepens  interest ;  for  it  is  but  common  sense  application 
of  a  habit  to  slightly  novel  conditions. 

4.  It  keeps  constantly  in  view  the  value  of  the  Image-imaging  quantity  and  quantity 
relations. 

5.  The  important  idea  of  "  balance  or  equation  "  is  frequently  stated  —  made  famil- 
iar to  the  pupil  —  for  the  insight  which  it  gives  into  the  problem  and  its  solution. 

6.  There  is  constant  insistence  on  the  clear  apprehension  and  statement  of  the  ele- 
ments of  the  question.    This  is  fundamental  in  the  intelligent  handling  of  problems. 

7.  From  the  gradual  psychological  development  of  the  subject,  the  method  is  given 
in  the  presentation  of  the  matter.  The  teacher  has  not  to  trouble  himself  with  books 
and  articles  on  methods  and  devices.  There  is  no  divorce  between  matter  and  method. 
Every  principle  and  process  follows  as  the  natural  movement  of  the  mind  demands. 
The  best  methods  are  followed  by  the  best  results. 

8.  Number  concepts  are  of  gradual  growth.  The  book  is  constructed  to  promote 
in  the  best  way  this  normal  growth.  To  secure  this  with  the  greatest  certainty  and 
economy,  all  the  questions  and  problems  in  the  book  are  original. 


THE   MACMILLAN   COMPANY 

NEW  YORK         CHICAGO         BOSTON  SAN  FRANCISCO  ATLANTA 


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